A001844 -id:A001844 - cp's OEIS frontend

A001653 Numbers k such that 2*k^2 - 1 is a square.

1, 5, 29, 169, 985, 5741, 33461, 195025, 1136689, 6625109, 38613965, 225058681, 1311738121, 7645370045, 44560482149, 259717522849, 1513744654945, 8822750406821, 51422757785981, 299713796309065, 1746860020068409, 10181446324101389, 59341817924539925

Offset: 1

N. J. A. Sloane

Consider all Pythagorean triples (X,X+1,Z) ordered by increasing Z; sequence gives Z values.
The defining equation is X^2 + (X+1)^2 = Z^2, which when doubled gives 2Z^2 = (2X+1)^2 + 1. So the sequence gives Z's such that 2Z^2 = odd square + 1 (A069894).
(x,y) = (a(n), a(n+1)) are the solutions with x < y of x/(yz) + y/(xz) + z/(xy)=3 with z=2. - Floor van Lamoen, Nov 29 2001
Consequently the sum n^2*(2n^2 - 1) of the first n odd cubes (A002593) is also a square. - Lekraj Beedassy, Jun 05 2002
Numbers n such that 2*n^2 = ceiling(sqrt(2)*n*floor(sqrt(2)*n)). - Benoit Cloitre, May 10 2003
Also, number of domino tilings in S_5 X P_2n. - Ralf Stephan, Mar 30 2004. Here S_5 is the star graph on 5 vertices with the edges {1,2}, {1,3}, {1,4}, {1,5}.
If x is in the sequence then so is x*(8*x^2-3). - James R. Buddenhagen, Jan 13 2005
In general, Sum_{k=0..n} binomial(2n-k,k)j^(n-k) = (-1)^n*U(2n,i*sqrt(j)/2), i=sqrt(-1). - Paul Barry, Mar 13 2005
a(n) = L(n,6), where L is defined as in A108299; see also A002315 for L(n,-6). - Reinhard Zumkeller, Jun 01 2005
Define a T-circle to be a first-quadrant circle with integral radius that is tangent to the x- and y-axes. Such a circle has coordinates equal to its radius. Let C(0) be the T-circle with radius 1. Then for n  >0, define C(n) to be the largest T-circle that intersects C(n-1). C(n) has radius a(n) and the coordinates of its points of intersection with C(n-1) are A001108(n) and A055997(n). Cf. A001109. - Charlie Marion, Sep 14 2005
Number of 01-avoiding words of length n on alphabet {0,1,2,3,4,5} which do not end in 0. - Tanya Khovanova, Jan 10 2007
The lower principal convergents to 2^(1/2), beginning with 1/1, 7/5, 41/29, 239/169, comprise a strictly increasing sequence; numerators = A002315 and denominators = {a(n)}. - Clark Kimberling, Aug 26 2008
Apparently Ljunggren shows that 169 is the last square term.
If (p,q) is a solution of the Diophantine equation: X^2 + (X+1)^2 = Y^2 then (p+q) or (p+q+1) are perfect squares. If (p,q) is a solution of the Diophantine equation: X^2 + (X+1)^2 = Y^2 then (p+q) or (p+q)/8 are perfect squares. If (p,q) and (r,s) are two consecutive solutions of the Diophantine equation: X^2 + (X+1)^2 = Y^2 with p < r then s-r = p+q+1. - Mohamed Bouhamida, Aug 29 2009
If (p,q) and (r,s) are two consecutive solutions of the Diophantine equation: X^2 + (X + 1)^2 = Y^2 with p < r then r = 3p+2q+1 and s = 4p+3q+2. - Mohamed Bouhamida, Sep 02 2009
Equals INVERT transform of A005054: (1, 4, 20, 100, 500, 2500, ...) and INVERTi transform of A122074: (1, 6, 40, 268, 1796, ...). - Gary W. Adamson, Jul 22 2010
a(n) is the number of compositions of n when there are 5 types of 1 and 4 types of other natural numbers. - Milan Janjic, Aug 13 2010
The remainder after division of a(n) by a(k) appears to belong to a periodic sequence: 1, 5, ..., a(k-1), 0, a(k)-a(k-1), ..., a(k)-1, a(k)-1, ..., a(k)-a(k-1), 0, a(k-1), ..., 5, 1. See Bouhamida's Sep 01 2009 comment. - Charlie Marion, May 02 2011
Apart from initial 1: subsequence of A198389, see also A198385. - Reinhard Zumkeller, Oct 25 2011
(a(n+1), 2*b(n+1)) and (a(n+2), 2*b(n+1)), n >= 0, with b(n):= A001109(n), give the (u(2*n), v(2*n)) and (u(2*n+1), v(2*n+1)) sequences, respectively, for Pythagorean triples (x,y,z), where x=|u^2-v^2|, y=2*u*v and z=u^2+v^2, with u odd and v even, which are generated from (u(0)=1, v(0)=2) by the substitution rule (u,v) -> (2*v+u,v) if u < v and (u,v) -> (u,2*u+v) if u > v. This leads to primitive triples because gcd(u,v) = 1 is respected. This corresponds to (primitive) Pythagorean triangles with |x-y|=1 (the catheti differ by one length unit). This (u,v) sequence starts with (1,2), (5,2), (5,12), (29,12), (29,70) ... - Wolfdieter Lang, Mar 06 2012
Area of the Fibonacci snowflake of order n. - José Luis Ramírez Ramírez, Dec 13 2012
Area of the 3-generalized Fibonacci snowflake of order n, n >= 3. - José Luis Ramírez Ramírez, Dec 13 2012
For the o.g.f. given by Johannes W. Meijer, Aug 01 2010, in the formula section see a comment under A077445. - Wolfdieter Lang, Jan 18 2013
Positive values of x (or y) satisfying x^2 - 6xy + y^2 + 4 = 0. - Colin Barker, Feb 04 2014
Length of period of the continued fraction expansion of a(n)*sqrt(2) is 1, the corresponding repeating value is A077444(n). - Ralf Stephan, Feb 20 2014
Positive values of x (or y) satisfying x^2 - 34xy + y^2 + 144 = 0. - Colin Barker, Mar 04 2014
The value of the hypotenuse in each triple of the Tree of primitive Pythagorean triples (cf. Wikipedia link) starting with root (3,4,5) and recursively selecting the central branch at each triple node of the tree. - Stuart E Anderson, Feb 05 2015
Positive integers z such that z^2 is a centered square number (A001844). - Colin Barker, Feb 12 2015
The aerated sequence (b(n)) n >= 1 = [1, 0, 5, 0, 29, 0, 169, 0, ...] is a fourth-order linear divisibility sequence; that is, if n | m then b(n) | b(m). It is the case P1 = 0, P2 = -8, Q = 1 of the 3-parameter family of divisibility sequences found by Williams and Guy. See A100047 for the connection with Chebyshev polynomials. - Peter Bala, Mar 25 2015
A002315(n-1)/a(n) is the closest rational approximation of sqrt(2) with a denominator not larger than a(n). These rational approximations together with those obtained from the sequences A001541 and A001542 give a complete set of closest rational approximations of sqrt(2) with restricted numerator or denominator. A002315(n-1)/a(n) < sqrt(2). - A.H.M. Smeets, May 28 2017
Equivalently, numbers x such that (x-1)*x/2 + x*(x+1)/2 = y^2 + (y+1)^2. y-values are listed in A001652. Example: for x=29 and y=20, 28*29/2 + 29*30/2 = 20^2 + 21^2. - Bruno Berselli, Mar 19 2018
From Wolfdieter Lang, Jun 13 2018: (Start)
(a(n), a(n+1)), with a(0):= 1, give all proper positive solutions m1 = m1(n) and m2 = m2(n), with m1 < m2 and n >= 0, of the Markoff triple (m, m1, m2) (see A002559) for m = 2, i.e., m1^2 - 6*m1*m2 + m2^2 = -4. Hence the unique Markoff triple with largest value m = 2 is (1, 1, 2) (for general m from A002559 this is the famous uniqueness conjecture).
For X = m2 - m1 and Y = m2 this becomes the reduced indefinite quadratic form representation X^2 + 4*X*Y - 4*Y^2 = -4, with discriminant 32, and the only proper fundamental solution (X(0), Y(0)) = (0, 1). For all nonnegative proper (X(n), Y(n)) solutions see (A005319(n) = a(n+1) - a(n), a(n+1)), for n >= 0. (End)
Each Pell(2*k+1) = a(k+1) number with k >= 3 appears as largest number of an ordered Markoff (Markov) triple [x, y, m] with smallest value x = 2 as [2, Pell(2*k-1), Pell(2*k+1)]. This known result follows also from all positive proper solutions of the Pell equation q^2 - 2*m^2 = -1 which are q = q(k) = A002315(k)  and m = m(k) = Pell(2*k+1), for k >= 0. y = y(k) = m(k) - 2*q(k) = Pell(2*k-1), with Pell(-1) = 1. The k = 0 and 1 cases do not satisfy x=2 <= y(k) <= m(k). The numbers 1 and 5 appear also as largest Markoff triple members because they are also Fibonacci numbers, and for these triples x=1. - Wolfdieter Lang, Jul 11 2018
All of the positive integer solutions of a*b+1=x^2, a*c+1=y^2, b*c+1=z^2, x+z=2*y, 0 < a < b < c are given by a=A001542(n), b=A005319(n), c=A001542(n+1), x=A001541(n), y=a(n+1), z=A002315(n) with 0 < n. - Michael Somos, Jun 26 2022

			From _Muniru A Asiru_, Mar 19 2018: (Start)
For k=1, 2*1^2 - 1 = 2 - 1 = 1 = 1^2.
For k=5, 2*5^2 - 1 = 50 - 1 = 49 = 7^2.
For k=29, 2*29^2 - 1 = 1682 - 1 = 1681 = 41^2.
... (End)
G.f. = x + 5*x^2 + 29*x^3 + 169*x^4 + 985*x^5 + 5741*x^6 + ... - _Michael Somos_, Jun 26 2022

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P.-F. Teilhet, Query 2376, L'Intermédiaire des Mathématiciens, 11 (1904), 138-139. - N. J. A. Sloane, Mar 08 2022
David Wells, The Penguin Dictionary of Curious and Interesting Numbers (Rev. ed. 1997), p. 91.

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Index entries for two-way infinite sequences
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (6,-1).

Other two sides are A001652, A046090.
Cf. A001519, A001109, A005054, A122074, A056220, A056869 (subset of primes).
Cf. A000217, A000290, A002315, A002559, A005319, A069894.
Row 6 of array A094954.
Row 1 of array A188647.
Cf. similar sequences listed in A238379.

a:=[1,5];; for n in [3..25] do a[n]:=6*a[n-1]-a[n-2]; od; a; # Muniru A Asiru, Mar 19 2018

a001653 n = a001653_list !! n
a001653_list = 1 : 5 : zipWith (-) (map (* 6) $ tail a001653_list) a001653_list
-- Reinhard Zumkeller, May 07 2013

I:=[1,5]; [n le 2 select I[n] else 6*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Feb 22 2014

a[0]:=1: a[1]:=5: for n from 2 to 26 do a[n]:=6*a[n-1]-a[n-2] od: seq(a[n], n=0..20); # Zerinvary Lajos, Jul 26 2006
A001653:=-(-1+5*z)/(z**2-6*z+1); # Conjectured (correctly) by Simon Plouffe in his 1992 dissertation; gives sequence except for one of the leading 1's

LinearRecurrence[{6,-1}, {1,5}, 40] (* Harvey P. Dale, Jul 12 2011 *)
a[ n_] := -(-1)^n ChebyshevU[2 n - 2, I]; (* Michael Somos, Jul 22 2018 *)
Numerator[{1} ~Join~
Table[FromContinuedFraction[Flatten[Table[{1, 4}, n]]], {n, 1, 40}]]; (* Greg Dresden, Sep 10 2019 *)

{a(n) = subst(poltchebi(n-1) + poltchebi(n), x, 3)/4}; /* Michael Somos, Nov 02 2002 */

a(n)=([5,2;2,1]^(n-1))[1,1] \\ Lambert Klasen (lambert.klasen(AT)gmx.de), corrected by Eric Chen, Jun 14 2018

{a(n) = -(-1)^n * polchebyshev(2*n-2, 2, I)}; /* Michael Somos, Jun 26 2022 */

G.f.: x*(1-x)/(1-6*x+x^2).
a(n) = 6*a(n-1) - a(n-2) with a(1)=1, a(2)=5.
4*a(n) = A077445(n).
Can be extended backwards by a(-n+1) = a(n).
a(n) = sqrt((A002315(n)^2 + 1)/2). [Inserted by N. J. A. Sloane, May 08 2000]
a(n+1) = S(n, 6)-S(n-1, 6), n>=0, with S(n, 6) = A001109(n+1), S(-2, 6) := -1. S(n, x)=U(n, x/2) are Chebyshev's polynomials of the second kind. Cf. triangle A049310. a(n+1) = T(2*n+1, sqrt(2))/sqrt(2), n>=0, with T(n, x) Chebyshev's polynomials of the first kind. [Offset corrected by Wolfdieter Lang, Mar 06 2012]
a(n) = A000129(2n+1). - Ira M. Gessel, Sep 27 2002
a(n) ~ (1/4)*sqrt(2)*(sqrt(2) + 1)^(2*n+1). - Joe Keane (jgk(AT)jgk.org), May 15 2002
a(n) = (((3 + 2*sqrt(2))^(n+1) - (3 - 2*sqrt(2))^(n+1)) - ((3 + 2*sqrt(2))^n - (3 - 2*sqrt(2))^n)) / (4*sqrt(2)). Limit_{n->infinity} a(n)/a(n-1) = 3 + 2*sqrt(2). - Gregory V. Richardson, Oct 12 2002
Let q(n, x) = Sum_{i=0..n} x^(n-i)*binomial(2*n-i, i); then q(n, 4) = a(n). - Benoit Cloitre, Nov 10 2002
For n and j >= 1, Sum_{k=0..j} a(k)*a(n) - Sum_{k=0..j-1} a(k)*a(n-1) = A001109(j+1)*a(n) - A001109(j)*a(n-1) = a(n+j); e.g., (1+5+29)*5 - (1+5)*1=169. - Charlie Marion, Jul 07 2003
From Charlie Marion, Jul 16 2003: (Start)
For n >= k >= 0, a(n)^2 = a(n+k)*a(n-k) - A084703(k)^2; e.g., 169^2 = 5741*5 - 144.
For n > 0, a(n) ^2 - a(n-1)^2 = 4*Sum_{k=0..2*n-1} a(k) = 4*A001109(2n); e.g., 985^2 - 169^2 = 4*(1 + 5 + 29 + ... + 195025) = 4*235416.
Sum_{k=0..n} ((-1)^(n-k)*a(k)) = A079291(n+1); e.g., -1 + 5 - 29 + 169 = 144.
A001652(n) + A046090(n) - a(n) = A001542(n); e.g., 119 + 120 - 169 = 70.
(End)
Sum_{k=0...n} ((2k+1)*a(n-k)) = A001333(n+1)^2 - (1 + (-1)^(n+1))/2; e.g., 1*169 + 3*29 + 5*5 + 7*1 = 288 = 17^2 - 1; 1*29 + 3*5 + 5*1 = 49 = 7^2. - Charlie Marion, Jul 18 2003
Sum_{k=0...n} a(k)*a(n) = Sum_{k=0..n} a(2k) and Sum_{k=0..n} a(k)*a(n+1) = Sum_{k=0..n} a(2k+1); e.g., (1+5+29)*29 = 1+29+985 and (1+5+29)*169 = 5+169+5741. - Charlie Marion, Sep 22 2003
For n >= 3, a_{n} = 7(a_{n-1} - a_{n-2}) + a_{n-3}, with a_1 = 1, a_2 = 5 and a_3 = 29. a(n) = ((-1+2^(1/2))/2^(3/2))*(3 - 2^(3/2))^n + ((1+2^(1/2))/2^(3/2))*(3 + 2^(3/2))^n. - Antonio Alberto Olivares, Oct 13 2003
Let a(n) = A001652(n), b(n) = A046090(n) and c(n) = this sequence. Then for k > j, c(i)*(c(k) - c(j)) = a(k+i) + ... + a(i+j+1) + a(k-i-1) + ... + a(j-i) + k - j. For n < 0, a(n) = -b(-n-1). Also a(n)*a(n+2k+1) + b(n)*b(n+2k+1) + c(n)*c(n+2k+1) = (a(n+k+1) - a(n+k))^2; a(n)*a(n+2k) + b(n)*b(n+2k) + c(n)*c(n+2k) = 2*c(n+k)^2. - Charlie Marion, Jul 01 2003
Let a(n) = A001652(n), b(n) = A046090(n) and c(n) = this sequence. Then for n > 0, a(n)*b(n)*c(n)/(a(n)+b(n)+c(n)) = Sum_{k=0..n} c(2*k+1); e.g., 20*21*29/(20+21+29) = 5+169 = 174; a(n)*b(n)*c(n)/(a(n-1)+b(n-1)+c(n-1)) = Sum_{k=0..n} c(2*k); e.g., 119*120*169/(20+21+29) = 1+29+985+33461 = 34476. - Charlie Marion, Dec 01 2003
Also solutions x > 0 of the equation floor(x*r*floor(x/r))==floor(x/r*floor(x*r)) where r=1+sqrt(2). - Benoit Cloitre, Feb 15 2004
a(n)*a(n+3) = 24 + a(n+1)*a(n+2). - Ralf Stephan, May 29 2004
For n >= k, a(n)*a(n+2*k+1) - a(n+k)*a(n+k+1) = a(k)^2-1; e.g., 29*195025-985*5741 = 840 = 29^2-1; 1*169-5*29 = 24 = 5^2-1; a(n)*a(n+2*k)-a(n+k)^2 = A001542(k)^2; e.g., 169*195025-5741^2 = 144 = 12^2; 1*29-5^2 = 4 = 2^2. - Charlie Marion Jun 02 2004
For all k, a(n) is a factor of a((2n+1)*k+n). a((2*n+1)*k+n) = a(n)*(Sum_{j=0..k-1} (-1)^j*(a((2*n+1)*(k-j)) + a((2*n+1)*(k-j)-1))+(-1)^k); e.g., 195025 = 5*(33461+5741-169-29+1); 7645370045 = 169*(6625109+1136689-1).- Charlie Marion, Jun 04 2004
a(n) = Sum_{k=0..n} binomial(n+k, 2*k)4^k. - Paul Barry, Aug 30 2004 [offset 0]
a(n) = Sum_{k=0..n} binomial(2*n+1, 2*k+1)*2^k. - Paul Barry, Sep 30 2004 [offset 0]
For n < k, a(n)*A001541(k) = A011900(n+k)+A053141(k-n-1); e.g., 5*99 = 495 = 493+2. For n >= k, a(n)*A001541(k) = A011900(n+k)+A053141(n-k); e.g., 29*3 = 87 = 85+2. - Charlie Marion, Oct 18 2004
a(n) = (-1)^n*U(2*n, i*sqrt(4)/2) = (-1)^n*U(2*n, i), U(n, x) Chebyshev polynomial of second kind, i=sqrt(-1). - Paul Barry, Mar 13 2005 [offset 0]
a(n) = Pell(2*n+1) = Pell(n)^2 + Pell(n+1)^2. - Paul Barry, Jul 18 2005 [offset 0]
a(n)*a(n+k) = A000129(k)^2 + A000129(2n+k+1)^2; e.g., 29*5741 = 12^2+169^2. - Charlie Marion, Aug 02 2005
Let a(n)*a(n+k) = x. Then 2*x^2-A001541(k)*x+A001109(k)^2 = A001109(2*n+k+1)^2; e.g., let x=29*985; then 2x^2-17x+6^2 = 40391^2; cf. A076218. - Charlie Marion, Aug 02 2005
With a=3+2*sqrt(2), b=3-2*sqrt(2): a(n) = (a^((2n+1)/2)+b^((2n+1)/2))/(2*sqrt(2)). a(n) = A001109(n+1)-A001109(n). - Mario Catalani (mario.catalani(AT)unito.it), Mar 31 2003
If k is in the sequence, then the next term is floor(k*(3+2*sqrt(2))). - Lekraj Beedassy, Jul 19 2005
a(n) = Jacobi_P(n,-1/2,1/2,3)/Jacobi_P(n,-1/2,1/2,1). - Paul Barry, Feb 03 2006 [offset 0]
a(n) = Sum_{k=0..n} Sum_{j=0..n-k} C(n,j)*C(n-j,k)*Pell(n-j+1), where Pell = A000129. - Paul Barry, May 19 2006 [offset 0]
a(n) = round(sqrt(A002315(n)^2/2)). - Lekraj Beedassy, Jul 15 2006
a(n) = A079291(n) + A079291(n+1). - Lekraj Beedassy, Aug 14 2006
a(n+1) = 3*a(n) + sqrt(8*a(n)^2-4), a(1)=1. - Richard Choulet, Sep 18 2007
6*a(n)*a(n+1) = a(n)^2+a(n+1)^2+4; e.g., 6*5*29 = 29^2+5^2+4; 6*169*985 = 169^2+985^2+4. - Charlie Marion, Oct 07 2007
2*A001541(k)*a(n)*a(n+k) = a(n)^2+a(n+k)^2+A001542(k)^2; e.g., 2*3*5*29 = 5^2+29^2+2^2; 2*99*29*5741 = 2*99*29*5741=29^2+5741^2+70^2. - Charlie Marion, Oct 12 2007
[a(n), A001109(n)] = [1,4; 1,5]^n * [1,0]. - Gary W. Adamson, Mar 21 2008
From Charlie Marion, Apr 10 2009: (Start)
In general, for n >= k, a(n+k) = 2*A001541(k)*a(n)-a(n-k);
e.g., a(n+0) = 2*1*a(n)-a(n); a(n+1) = 6*a(n)-a(n-1); a(6+0) = 33461 = 2*33461-33461; a(5+1) = 33461 = 6*5741-985; a(4+2) = 33461 = 34*985-29; a(3+3) = 33461 = 198*169-1.
(End)
G.f.: sqrt(x)*tan(4*arctan(sqrt(x)))/4. - Johannes W. Meijer, Aug 01 2010
Given k = (sqrt(2)+1)^2 = 3+2*sqrt(2) and a(0)=1, then a(n) = a(n-1)*k-((k-1)/(k^n)). - Charles L. Hohn, Mar 06 2011
Given k = (sqrt(2)+1)^2 = 3+2*sqrt(2) and a(0)=1, then a(n) = (k^n)+(k^(-n))-a(n-1) = A003499(n) - a(n-1). - Charles L. Hohn, Apr 04 2011
Let T(n) be the n-th triangular number; then, for n > 0, T(a(n)) + A001109(n-1) = A046090(n)^2.  See also A046090. - Charlie Marion, Apr 25 2011
For k > 0, a(n+2*k-1) - a(n) = 4*A001109(n+k-1)*A002315(k-1); a(n+2*k) - a(n) = 4*A001109(k)*A002315(n+k-1). - Charlie Marion, Jan 06 2012
a(k+j+1) = (A001541(k)*A001541(j) + A002315(k)*A002315(j))/2. - Charlie Marion, Jun 25 2012
a(n)^2 = 2*A182435(n)*(A182435(n)-1)+1. - Bruno Berselli, Oct 23 2012
a(n) = A143608(n-1)*A143608(n) + 1 = A182190(n-1)+1. - Charlie Marion, Dec 11 2012
G.f.: G(0)*(1-x)/(2-6*x), where G(k) = 1 + 1/(1 - x*(8*k-9)/( x*(8*k-1) - 3/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 12 2013
a(n+1) = 4*A001652(n) + 3*a(n) + 2 [Mohamed Bouhamida's 2009 (p,q)(r,s) comment above rewritten]. - Hermann Stamm-Wilbrandt, Jul 27 2014
a(n)^2 = A001652(n-1)^2 + (A001652(n-1)+1)^2. - Hermann Stamm-Wilbrandt, Aug 31 2014
Sum_{n >= 2} 1/( a(n) - 1/a(n) ) = 1/4. - Peter Bala, Mar 25 2015
a(n) = Sum_{k=0..n} binomial(n,k) * 3^(n-k) * 2^k * 2^floor(k/2). - David Pasino, Jul 09 2016
E.g.f.: (sqrt(2)*sinh(2*sqrt(2)*x) + 2*cosh(2*sqrt(2)*x))*exp(3*x)/2. - Ilya Gutkovskiy, Jul 09 2016
a(n+2) = (a(n+1)^2 + 4)/a(n). - Vladimir M. Zarubin, Sep 06 2016
a(n) = 2*A053141(n)+1. - R. J. Mathar, Aug 16 2019
For n>1, a(n) is the numerator of the continued fraction [1,4,1,4,...,1,4] with (n-1) repetitions of 1,4. For the denominators see A005319. - Greg Dresden, Sep 10 2019
a(n) = round(((2+sqrt(2))*(3+2*sqrt(2))^(n-1))/4). - Paul Weisenhorn, May 23 2020
a(n+1) = Sum_{k >= n} binomial(2*k,2*n)*(1/2)^(k+1). Cf. A102591. - Peter Bala, Nov 29 2021
a(n+1) = 3*a(n) + A077444(n). - César Aguilera, Jul 13 2023

Additional comments from Wolfdieter Lang, Feb 10 2000
Better description from Harvey P. Dale, Jan 15 2002
Edited by N. J. A. Sloane, Nov 02 2002

A008586 Multiples of 4.

Original entry on oeis.org

0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 128, 132, 136, 140, 144, 148, 152, 156, 160, 164, 168, 172, 176, 180, 184, 188, 192, 196, 200, 204, 208, 212, 216, 220, 224, 228

Offset: 0

N. J. A. Sloane

Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0( 14 ).
A000466(n), a(n) and A053755(n) are Pythagorean triples. - Zak Seidov, Jan 16 2007
If X is an n-set and Y and Z disjoint 2-subsets of X then a(n-3) is equal to the number of 3-subsets of X intersecting both Y and Z. - Milan Janjic, Aug 26 2007
Number of n-permutations (n>=1) of 5 objects u, v, z, x, y with repetition allowed, containing n-1 u's. Example: if n=1 then n-1 = zero (0) u, a(1)=4 because we have v, z, x, y. If n=2 then n-1 = one (1) u, a(2)=8 because we have vu, zu, xu, yu, uv, uz, ux, uy. A038231 formatted as a triangular array: diagonal: 4, 8, 12, 16, 20, 24, 28, 32, ... - Zerinvary Lajos, Aug 06 2008
For n > 0: numbers having more even than odd divisors: A048272(a(n)) < 0. - Reinhard Zumkeller, Jan 21 2012
A214546(a(n)) < 0 for n > 0. - Reinhard Zumkeller, Jul 20 2012
A090418(a(n)) = 0 for n > 0. - Reinhard Zumkeller, Aug 06 2012
Terms are the differences of consecutive centered square numbers (A001844). - Mihir Mathur, Apr 02 2013
a(n)*Pi = nonnegative zeros of the cycloid generated by a circle of radius 2 rolling along the positive x-axis from zero. - Wesley Ivan Hurt, Jul 01 2013
Apart from the initial term, number of vertices of minimal path on an n-dimensional cubic lattice (n>1) of side length 2, until a self-avoiding walk gets stuck. A004767 + 1. - Matthew Lehman, Dec 23 2013
The number of orbits of Aut(Z^7) as function of the infinity norm n of the representative lattice point of the orbit, when the cardinality of the orbit is equal to 2688. - Philippe A.J.G. Chevalier, Dec 29 2015
First differences of A001844. - Robert Price, May 13 2016
Numbers k such that Fibonacci(k) is a multiple of 3 (A033888). - Bruno Berselli, Oct 17 2017

T. D. Noe, Table of n, a(n) for n = 0..1000
Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 3.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 316 [Broken link]
Milan Janjic, Two Enumerative Functions
Tanya Khovanova, Recursive Sequences
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014, 2015.
William A. Stein, The modular forms database
Eric Weisstein's World of Mathematics, Doubly Even Number
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A000290, A000466, A001844, A004767, A008574, A030308, A033888, A035008, A038231, A048272, A053755, A090418, A214546.

Number of orbits of Aut(Z^7) as function of the infinity norm A000579, A154286, A102860, A002412, A045943, A115067, A008585, A005843, A001477, A000217.

a008586 = (* 4)
a008586_list = [0, 4 ..]  -- Reinhard Zumkeller, May 13 2014

A008586:=n->4*n; seq(A008586(n), n=0..100); # Wesley Ivan Hurt, Feb 24 2014

Range[0, 500, 4] (* Vladimir Joseph Stephan Orlovsky, May 26 2011 *)

a(n)=n<<2 \\ Charles R Greathouse IV, Oct 17 2011

a(n) = A008574(n), n>0. - R. J. Mathar, Oct 28 2008
a(n) = Sum_{k>=0} A030308(n,k)*2^(k+2). - Philippe Deléham, Oct 17 2011
a(n+1) = A000290(n+2) - A000290(n). - Philippe Deléham, Mar 31 2013
G.f.: 4*x/(1-x)^2. - David Wilding, Jun 21 2014
E.g.f.: 4*x*exp(x). - Stefano Spezia, May 18 2021

A046092 4 times triangular numbers: a(n) = 2n(n+1).

Original entry on oeis.org

0, 4, 12, 24, 40, 60, 84, 112, 144, 180, 220, 264, 312, 364, 420, 480, 544, 612, 684, 760, 840, 924, 1012, 1104, 1200, 1300, 1404, 1512, 1624, 1740, 1860, 1984, 2112, 2244, 2380, 2520, 2664, 2812, 2964, 3120, 3280, 3444, 3612, 3784, 3960, 4140, 4324

Offset: 0

N. J. A. Sloane and Eric W. Weisstein

Consider all Pythagorean triples (X,Y,Z=Y+1) ordered by increasing Z; sequence gives Y values. X values are 1, 3, 5, 7, 9, ... (A005408), Z values are A001844.
In the triple (X, Y, Z) we have X^2=Y+Z. Actually, the triple is given by {x, (x^2 -+ 1)/2}, where x runs over the odd numbers (A005408) and x^2 over the odd squares (A016754). - Lekraj Beedassy, Jun 11 2004
a(n) is the number of edges in n X n square grid with all horizontal and vertical segments filled in. - Asher Auel, Jan 12 2000 [Corrected by Felix Huber, Apr 09 2024]
a(n) is the only number satisfying an inequality related to zeta(2) and zeta(3): Sum_{i>a(n)+1} 1/i^2 < Sum_{i>n} 1/i^3 < Sum_{i>a(n)} 1/i^2. - Benoit Cloitre, Nov 02 2001
Number of right triangles made from vertices of a regular n-gon when n is even. - Sen-Peng Eu, Apr 05 2001
Number of ways to change two non-identical letters in the word aabbccdd..., where there are n type of letters. - Zerinvary Lajos, Feb 15 2005
a(n) is the number of (n-1)-dimensional sides of an (n+1)-dimensional hypercube (e.g., squares have 4 corners, cubes have 12 edges, etc.). - Freek van Walderveen (freek_is(AT)vanwal.nl), Nov 11 2005
From Nikolaos Diamantis (nikos7am(AT)yahoo.com), May 23 2006: (Start)
Consider a triangle, a pentagon, a heptagon, ..., a k-gon where k is odd. We label a triangle with n=1, a pentagon with n=2, ..., a k-gon with n = floor(k/2). Imagine a player standing at each vertex of the k-gon.
Initially there are 2 frisbees, one held by each of two neighboring players. Every time they throw the frisbee to one of their two nearest neighbors with equal probability. Then a(n) gives the average number of steps needed so that the frisbees meet.
I verified this by simulating the processes with a computer program. For example, a(2) = 12 because in a pentagon that's the expected number of trials we need to perform. That is an exercise in Concrete Mathematics and it can be done using generating functions. (End)
A diagonal of A059056. - Zerinvary Lajos, Jun 18 2007
If X_1,...,X_n is a partition of a 2n-set X into 2-blocks then a(n-1) is equal to the number of 2-subsets of X containing none of X_i, (i=1,...,n). - Milan Janjic, Jul 16 2007
X values of solutions to the equation 2*X^3 + X^2 = Y^2. To find Y values: b(n) = 2n(n+1)(2n+1). - Mohamed Bouhamida, Nov 06 2007
Number of (n+1)-permutations of 3 objects u,v,w, with repetition allowed, containing n-1 u's. Example: a(1)=4 because we have vv, vw, wv and ww; a(2)=12 because we can place u in each of the previous four 2-permutations either in front, or in the middle, or at the end. - Zerinvary Lajos, Dec 27 2007
Sequence found by reading the line from 0, in the direction 0, 4, ... and the same line from 0, in the direction 0, 12, ..., in the square spiral whose vertices are the triangular numbers A000217. - Omar E. Pol, May 03 2008
a(n) is also the least weight of self-conjugate partitions having n different even parts. - Augustine O. Munagi, Dec 18 2008
From Peter Luschny, Jul 12 2009: (Start)
The general formula for alternating sums of powers of even integers is in terms of the Swiss-Knife polynomials P(n,x) A153641 (P(n,1)-(-1)^k P(n,2k+1))/2. Here n=2, thus
a(k) = |(P(2,1) - (-1)^k*P(2,2k+1))/2|. (End)
The sum of squares of n+1 consecutive numbers between a(n)-n and a(n) inclusive equals the sum of squares of n consecutive numbers following a(n). For example, for n = 2, a(2) = 12, and the corresponding equation is 10^2 + 11^2 + 12^2 = 13^2 + 14^2. - Tanya Khovanova, Jul 20 2009
Number of roots in the root system of type D_{n+1} (for n>2). - Tom Edgar, Nov 05 2013
Draw n ellipses in the plane (n>0), any 2 meeting in 4 points; sequence gives number of intersections of these ellipses (cf. A051890, A001844); a(n) = A051890(n+1) - 2 = A001844(n) - 1. - Jaroslav Krizek, Dec 27 2013
a(n) appears also as the second member of the quartet [p0(n), a(n), p2(n), p3(n)] of the square of [n, n+1, n+2, n+3] in the Clifford algebra Cl_2 for n >= 0. p0(n) = -A147973(n+3), p2(n) = A054000(n+1) and p3(n) = A139570(n). See a comment on A147973, also with a reference. - Wolfdieter Lang, Oct 15 2014
a(n) appears also as the third and fourth member of the quartet [p0(n), p0(n), a(n), a(n)] of the square of [n, n, n+1, n+1] in the Clifford algebra Cl_2 for n >= 0. p0(n) = A001105(n). - Wolfdieter Lang, Oct 16 2014
Consider two equal rectangles composed of unit squares. Then surround the 1st rectangle with 1-unit-wide layers to build larger rectangles, and surround the 2nd rectangle just to hide the previous layers. If r(n) and h(n) are the number of unit squares needed for n layers in the 1st case and the 2nd case, then for all rectangles, we have a(n) = r(n) - h(n) for n>=1. - Michel Marcus, Sep 28 2015
When greater than 4, a(n) is the perimeter of a Pythagorean triangle with an even short leg 2*n. - Agola Kisira Odero, Apr 26 2016
Also the number of minimum connected dominating sets in the (n+1)-cocktail party graph. - Eric W. Weisstein, Jun 29 2017
a(n+1) is the harmonic mean of A000384(n+2) and A014105(n+1). - Bob Andriesse, Apr 27 2019
Consider a circular cake from which wedges of equal center angle c are cut out in clockwise succession and turned around so that the bottom comes to the top. This goes on until the cake shows its initial surface again. An interesting case occurs if 360°/c is not an integer. Then, with n = floor(360°/c), the number of wedges which have to be cut out and turned equals a(n). (For the number of cutting line segments see A005408.) - According to Peter Winkler's book "Mathematical Mind-Benders", which presents the problem and its solution (see Winkler, pp. 111, 115) the problem seems to be of French origin but little is known about its history. - Manfred Boergens, Apr 05 2022
a(n-3) is the maximum irregularity over all maximal 2-degenerate graphs with n vertices.  The extremal graphs are 2-stars (K_2 joined to n-2 independent vertices).  (The irregularity of a graph is the sum of the differences between the degrees over all edges of the graph.) - Allan Bickle, May 29 2023
Number of ways of placing a domino on a (n+1)X(n+1) board of squares. - R. J. Mathar, Apr 24 2024
The sequence terms are the exponents in the expansion of (1/(1 + x)) * Sum_{n >= 0} x^n * Product_{k = 1..n} (1 - x^(2*k-1))/(1 + x^(2*k+1)) = 1 - x^4 + x^12 - x^24 + x^40 - x^60 + - ... (Andrews and Berndt, Entry 9.3.3, p. 229). Cf. A153140. - Peter Bala, Feb 15 2025
Number of edges in an (n+1)-dimensional orthoplex. 2D orthoplexes (diamonds) have 4 edges, 3D orthoplexes (octahedrons) have 12 edges, 4D orthoplexes (16-cell) have 24 edges, and so on. - Aaron Franke, Mar 23 2025

			a(7)=112 because 112 = 2*7*(7+1).
The first few triples are (1,0,1), (3,4,5), (5,12,13), (7,24,25), ...
The first such partitions, corresponding to a(n)=1,2,3,4, are 2+2, 4+4+2+2, 6+6+4+4+2+2, 8+8+6+6+4+4+2+2. - _Augustine O. Munagi_, Dec 18 2008

George E. Andrews and Bruce C. Berndt, Ramanujan's Lost Notebook, Part I, Springer, 2005.
Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 3.
Albert H. Beiler, Recreations in the Theory of Numbers. New York: Dover, p. 125, 1964.
Ronald L. Graham, D. E. Knuth and Oren Patashnik, Concrete Mathematics, Reading, Massachusetts: Addison-Wesley, 1994.
Peter Winkler, Mathematical Mind-Benders, Wellesley, Massachusetts: A K Peters, 2007.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Allan Bickle and Zhongyuan Che, Irregularities of Maximal k-degenerate Graphs, Discrete Applied Math. 331 (2023) 70-87.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
H. J. Brothers, Pascal's Prism: Supplementary Material.
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, preprint, 2016.
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, Vol. 13, No. 4 (2017), Article #47.
Z. Janelidze, F. van Niekerk, and J. Viljoen, What is the maximal connected partial symmetry index of a connected graph of a given size?, arXiv:2502.00704 [math.CO], 2025. See p. 3.
Milan Janjic, Two Enumerative Functions
Ron Knott, Pythagorean Triples and Online Calculators
Tanya Khovanova, A Miracle Equation.
Augustine O. Munagi, Pairing conjugate partitions by residue classes, Discrete Math., 308 (2008), 2492-2501. [From _Augustine O. Munagi_, Dec 18 2008]
Enrique Navarrete and Daniel Orellana, Finding Prime Numbers as Fixed Points of Sequences, arXiv:1907.10023 [math.NT], 2019.
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.
Amelia Carolina Sparavigna, The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences, Politecnico di Torino, Italy (2019), [math.NT].
Amelia Carolina Sparavigna, Some Groupoids and their Representations by Means of Integer Sequences, International Journal of Sciences (2019) Vol. 8, No. 10.
Rusliansyah D. Suprijanto, Observation on Sums of Powers of Integers Divisible by Four, Applied Mathematical Sciences, Vol. 8, 2014, no. 45, 2219 - 2226.
Leo Tavares, Illustration: Diamond Rows
Herman Tulleken, Polyominoes 2.2: How they fit together, (2019).
Eric Weisstein's World of Mathematics, Aztec Diamond.
Eric Weisstein's World of Mathematics, Cocktail Party Graph.
Eric Weisstein's World of Mathematics, Connected Dominating Set.
Eric Weisstein's World of Mathematics, Gear Graph.
Eric Weisstein's World of Mathematics, Hamiltonian Path.
Eric Weisstein's World of Mathematics, Pythagorean Triple.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A045943, A028895, A002943, A054000, A000330, A007290, A002378, A033996, A124080, A028896, A049598, A005563, A000217, A033586, A085250.
Main diagonal of array in A001477.
Equals A033996/2. Cf. A001844. - Augustine O. Munagi, Dec 18 2008
Cf. A078371, A141530 (see Librandi's comment in A078371).
Cf. A097080, A001845.
Cf. similar sequences listed in A299645.
Cf. A005408.
Cf. A016754.
Cf. A002378, A046092, A028896 (irregularities of maximal k-degenerate graphs).

a046092 = (* 2) . a002378  -- Reinhard Zumkeller, Dec 15 2013

[2*n*(n+1): n in [0..50]]; // Vincenzo Librandi, Oct 04 2011

Table[2 n (n + 1), {n, 0, 50}] (* Stefan Steinerberger, Apr 03 2006 *)
LinearRecurrence[{3, -3, 1}, {0, 4, 12}, 50] (* Harvey P. Dale, Jul 25 2011 *)
4*Binomial[Range[50], 2] (* Harvey P. Dale, Jul 25 2011 *)

A046092(n):=2*n*(n+1)$
makelist(A046092(n),n,0,30); /* Martin Ettl, Nov 08 2012 */

a(n)=binomial(n+1,2)<<2 \\ Charles R Greathouse IV, Jun 10 2011

def A046092(n): return n*(n+1)<<1 # Chai Wah Wu, Mar 11 2025

a(n) = A100345(n+1, n-1) for n>0.
a(n) = 2*A002378(n) = 4*A000217(n). - Lekraj Beedassy, May 25 2004
a(n) = C(2n, 2) - n = 4*C(n, 2). - Zerinvary Lajos, Feb 15 2005
From Lekraj Beedassy, Jun 04 2006: (Start)
a(n) - a(n-1)=4*n.
Let k=a(n). Then a(n+1) = k + 2*(1 + sqrt(2k + 1)). (End)
Array read by rows: row n gives A033586(n), A085250(n+1). - Omar E. Pol, May 03 2008
O.g.f.:4*x/(1-x)^3; e.g.f.: exp(x)*(2*x^2+4*x). - Geoffrey Critzer, May 17 2009
From Stephen Crowley, Jul 26 2009: (Start)
a(n) = 1/int(-(x*n+x-1)*(step((-1+x*n)/n)-1)*n*step((x*n+x-1)/(n+1)),x=0..1) where step(x)=piecewise(x<0,0,0<=x,1) is the Heaviside step function.
Sum_{n>=1} 1/a(n) = 1/2. (End)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=0, a(1)=4, a(2)=12. - Harvey P. Dale, Jul 25 2011
For n > 0, a(n) = 1/(Integral_{x=0..Pi/2} (sin(x))^(2*n-1)*(cos(x))^3). - Francesco Daddi, Aug 02 2011
a(n) = A001844(n) - 1. - Omar E. Pol, Oct 03 2011
(a(n) - A000217(k))^2  = A000217(2n-k)*A000217(2n+1+k) - (A002378(n) - A000217(k)), for all k. See also A001105. - Charlie Marion, May 09 2013
From Ivan N. Ianakiev, Aug 30 2013: (Start)
a(n)*(2m+1)^2 + a(m) = a(n*(2m+1)+m), for any nonnegative integers n and m.
t(k)*a(n) + t(k-1)*a(n+1) = a((n+1)*(t(k)-t(k-1)-1)), where k>=2, n>=1, t(k)=A000217(k). (End)
a(n) = A245300(n,n). - Reinhard Zumkeller, Jul 17 2014
2*a(n)+1 = A016754(n) = A005408(n)^2, the odd squares. - M. F. Hasler, Oct 02 2014
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2) - 1/2 = A187832. - Ilya Gutkovskiy, Mar 16 2017
a(n) = lcm(2*n,2*n+2). - Enrique Navarrete, Aug 30 2017
a(n)*a(n+k) + k^2 = m^2 (a perfect square), n >= 1, k >= 0. - Ezhilarasu Velayutham, May 13 2019
From Amiram Eldar, Jan 29 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = cosh(Pi/2)/(Pi/2).
Product_{n>=1} (1 - 1/a(n)) = -2*cos(sqrt(3)*Pi/2)/Pi. (End)
a(n) = A016754(n) - A001844(n). - Leo Tavares, Sep 20 2022

A005448 Centered triangular numbers: a(n) = 3n(n-1)/2 + 1.

Original entry on oeis.org

1, 4, 10, 19, 31, 46, 64, 85, 109, 136, 166, 199, 235, 274, 316, 361, 409, 460, 514, 571, 631, 694, 760, 829, 901, 976, 1054, 1135, 1219, 1306, 1396, 1489, 1585, 1684, 1786, 1891, 1999, 2110, 2224, 2341, 2461, 2584, 2710, 2839, 2971, 3106, 3244, 3385, 3529

Offset: 1

N. J. A. Sloane, R. K. Guy, Dec 12 1974

These are Hogben's central polygonal numbers
2
.P
3 n
Also the sum of three consecutive triangular numbers (A000217); i.e., a(4) = 19 = T4 + T3 + T2 = 10 + 6 + 3. - Robert G. Wilson v, Apr 27 2001
For k>2, Sum_{n=1..k} a(n) gives the sum pertaining to the magic square of order k. E.g., Sum_{n=1..5} a(n) = 1 + 4 + 10 + 19 + 31 = 65. In general, Sum_{n=1..k} a(n) = k*(k^2 + 1)/2. - Amarnath Murthy, Dec 22 2001
Binomial transform of (1,3,3,0,0,0,...). - Paul Barry, Jul 01 2003
a(n) is the difference of two tetrahedral (or pyramidal) numbers: C(n+3,3) = (n+1)(n+2)(n+3)/6. a(n) = A000292(n) - A000292(n-3) = (n+1)(n+2)(n+3)/6 - (n-2)(n-1)(n)/6. - Alexander Adamchuk, May 20 2006
Partial sums are A006003(n) = n(n^2+1)/2. Finite differences are a(n+1) - a(n) = A008585(n) = 3n. - Alexander Adamchuk, Jun 03 2006
If X is an n-set and Y a fixed 3-subset of X then a(n-2) is equal to the number of 3-subsets of X intersecting Y. - Milan Janjic, Jul 30 2007
Equals (1, 2, 3, ...) convolved with (1, 2, 3, 3, 3, ...). a(4) = 19 = (1, 2, 3, 4) dot (3, 3, 2, 1) = (3 + 6 + 6 + 4). - Gary W. Adamson, May 01 2009
Equals the triangular numbers convolved with [1, 1, 1, 0, 0, 0, ...]. - Gary W. Adamson and Alexander R. Povolotsky, May 29 2009
a(n) is the number of triples (w,x,y) having all terms in {0,...,n} and min(w+x,x+y,y+w) = max(w,x,y). - Clark Kimberling, Jun 14 2012
a(n) = number of atoms at graph distance <= n from an atom in the graphite or graphene network (cf. A008486). - N. J. A. Sloane, Jan 06 2013
In 1826, Shiraishi gave a solution to the Diophantine equation a^3 + b^3 + c^3 = d^3 with b = a(n) for n > 1; see A226903. - Jonathan Sondow, Jun 22 2013
For n > 1, a(n) is the remainder of n^2 * (n-1)^2 mod (n^2 + (n-1)^2). - J. M. Bergot, Jun 27 2013
The equation A000578(x) - A000578(x-1) = A000217(y) - A000217(y-2) is satisfied by y=a(x). - Bruno Berselli, Feb 19 2014
A242357(a(n)) = n. - Reinhard Zumkeller, May 11 2014
A255437(a(n)) = 1. - Reinhard Zumkeller, Mar 23 2015
The first differences give A008486. a(n) seems to give the total number of triangles in the n-th generation of the six patterns of triangle expansion shown in the link. - Kival Ngaokrajang, Sep 12 2015
Number of binary shuffle squares of length 2n which contains exactly two 1's. - Bartlomiej Pawlik, Sep 07 2023
The digital root has period 3 (1, 4, 1) (A146325), the same digital root as the centered 12-gonal numbers, or centered dodecagonal numbers A003154(n). - Peter M. Chema, Dec 20 2023

			From _Seiichi Manyama_, Aug 12 2017: (Start)
a(1) = 1:
      *
     / \
    /   \
   /     \
  *-------*
.................................................
a(2) = 4:
            *
           / \
          /   \
         /     \
        *---*---*
           / \
      *   /   \   *
     / \ /     \ / \
    /   *-------*   \
   /     \     /     \
  *-------*   *-------*
.................................................
a(3) = 10:
                  *
                 / \
                /   \
               /     \
              *---*---*
                 / \
            *   /   \   *
           / \ /     \ / \
          /   *---*---*   \
         /     \ / \ /     \
        *---*---*   *---*---*
           / \ /     \ / \
      *   /   *---*---*   \   *
     / \ /     \ / \ /     \ / \
    /   *-------*   *-------*   \
   /     \     /     \     /     \
  *-------*   *-------*   *-------*
.................................................
a(4) = 19:
                        *
                       / \
                      /   \
                     /     \
                    *---*---*
                       / \
                  *   /   \   *
                 / \ /     \ / \
                /   *---*---*   \
               /     \ / \ /     \
              *---*---*   *---*---*
                 / \ /     \ / \
            *   /   \---*---*   \   *
           / \ /     \ / \ /     \ / \
          /   *---*---*   *---*---*   \
         /     \ / \ /     \ / \ /     \
        *---*---*   *---*---*   *---*---*
           / \ /     \ / \ /     \ / \
      *   /   *---*---*   *---*---*   \   *
     / \ /     \ / \ /     \ / \ /     \ / \
    /   *-------*   *-------*   *-------*   \
   /     \     /     \     /     \     /     \
  *-------*   *-------*   *-------*   *-------*
(End)

R. Reed, The Lemming Simulation Problem, Mathematics in School, 3 (#6, Nov. 1974), front cover and pp. 5-6.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
D. Bevan, D. Levin, P. Nugent, J. Pantone, and L. Pudwell, Pattern avoidance in forests of binary shrubs, arXiv:1510.08036 [math.CO], 2015.
Jarosław Grytczuk, Bartłomiej Pawlik, and Mariusz Pleszczyński, Variations on shuffle squares, arXiv:2308.13882 [math.CO], 2023. See p. 11.
F. Javier de Vega, On the parabolic partitions of a number, J. Alg., Num. Theor., and Appl. (2023) Vol. 61, No. 2, 135-169.
Guo-Niu Han, Enumeration of Standard Puzzles [Cached copy]
L. Hogben, Choice and Chance by Cardpack and Chessboard, Vol. 1, Max Parrish and Co, London, 1950, p. 22.
Milan Janjic, Two Enumerative Functions
Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.
Kival Ngaokrajang, Illustration of triangles expansion
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
R. Reed, The Lemming Simulation Problem, Mathematics in School, 3 (#6, Nov. 1974), front cover and pp. 5-6. [Scanned photocopy of pages 5, 6 only, with annotations by R. K. Guy and N. J. A. Sloane]
Leo Tavares, Illustration: Triple Triangles
B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.
Eric Weisstein's World of Mathematics, Centered Triangular Number
Julie Zhang, Noah A. Rosenberg, and Julia A. Palacios, The space of multifurcating ranked tree shapes: enumeration, lattice structure, and Markov chains, arXiv:2506.10856 [math.PR], 2025. See p. 33.
Index entries for sequences related to centered polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A000292, A001263, A001844, A002061, A003154, A006003 (partial sums), A008486, A008585 = first differences, A045943, A134482, A146325, A226903, A242357, A255437.

a005448 n = 3 * n * (n - 1) `div` 2 + 1
a005448_list = 1 : zipWith (+) a005448_list [3, 6 ..]
-- Reinhard Zumkeller, Jun 20 2013

I:=[1,4,10]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..60]]; // Vincenzo Librandi, Sep 13 2015

A005448 := n->(3*(n-1)^2+3*(n-1)+2)/2: seq(A005448(n), n=1..100);
A005448 := -(1+z+z**2)/(z-1)^3; # Simon Plouffe in his 1992 dissertation for offset 0

FoldList[#1 + #2 &, 1, 3 Range@ 50] (* Robert G. Wilson v, Feb 02 2011 *)
Join[{1,4},Total/@Partition[Accumulate[Range[50]],3,1]] (* Harvey P. Dale, Aug 17 2012 *)
LinearRecurrence[{3, -3, 1}, {1, 4, 10}, 50] (* Vincenzo Librandi, Sep 13 2015 *)
Table[ j! Coefficient[Series[Exp[x]*(1 + 3 x^2/2)-1, {x, 0, 20}], x, j], {j, 0, 20}] (* Nikolaos Pantelidis, Feb 07 2023 *)
3#+1&/@Accumulate[Range[0,50]] (* Harvey P. Dale, Nov 20 2024 *)

{a(n)=3*(n^2-n)/2+1} /* Michael Somos, Sep 23 2006 */

isok(n) = my(k=(2*n-2)/3, m); (n==1) || ((denominator(k)==1) && (m=sqrtint(k)) && (m*(m+1)==k)); \\ Michel Marcus, May 20 2020

Expansion of x*(1-x^3)/(1-x)^4.
a(n) = C(n+3, 3)-C(n, 3) = C(n, 0)+3*C(n, 1)+3*C(n, 2). - Paul Barry, Jul 01 2003
a(n) = 1 + Sum_{j=0..n-1} (3*j). - Xavier Acloque, Oct 25 2003
a(n) = A000217(n) + A000290(n-1) = (3*A016754(n) + 5)/8. - Lekraj Beedassy, Nov 05 2005
Euler transform of length 3 sequence [4, 0, -1]. - Michael Somos, Sep 23 2006
a(1-n) = a(n). - Michael Somos, Sep 23 2006
a(n) = binomial(n+1,n-1) + binomial(n,n-2) + binomial(n-1,n-3). - Zerinvary Lajos, Sep 03 2006
Row sums of triangle A134482. - Gary W. Adamson, Oct 27 2007
Narayana transform (A001263) * [1, 3, 0, 0, 0, ...]. - Gary W. Adamson, Dec 29 2007
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), a(1)=1, a(2)=4, a(3)=10. - Jaume Oliver Lafont, Dec 02 2008
a(n) = A000217(n-1)*3 + 1 = A045943(n-1) + 1. - Omar E. Pol, Dec 27 2008
a(n) = a(n-1) + 3*n-3. - Vincenzo Librandi, Nov 18 2010
Sum_{n>=1} 1/a(n) = A306324. - Ant King, Jun 12 2012
a(n) = 2*a(n-1) - a(n-2) + 3. - Ant King, Jun 12 2012
a(n) = A101321(3,n-1). - R. J. Mathar, Jul 28 2016
E.g.f.: -1 + (2 + 3*x^2)*exp(x)/2. - Ilya Gutkovskiy, Jul 28 2016
a(n) = A002061(n) + A000217(n-1). - Bruce J. Nicholson, Apr 20 2017
From Amiram Eldar, Jun 20 2020: (Start)
Sum_{n>=1} a(n)/n! = 5*e/2 - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 5/(2*e) - 1. (End)
a(n) = A000326(n) - n + 1. - Charlie Marion, Nov 21 2020

A005449 Second pentagonal numbers: a(n) = n(3n + 1)/2.

Original entry on oeis.org

0, 2, 7, 15, 26, 40, 57, 77, 100, 126, 155, 187, 222, 260, 301, 345, 392, 442, 495, 551, 610, 672, 737, 805, 876, 950, 1027, 1107, 1190, 1276, 1365, 1457, 1552, 1650, 1751, 1855, 1962, 2072, 2185, 2301, 2420, 2542, 2667, 2795, 2926, 3060, 3197, 3337, 3480

Offset: 0

N. J. A. Sloane

Number of edges in the join of the complete graph and the cycle graph, both of order n, K_n * C_n. - Roberto E. Martinez II, Jan 07 2002
Also number of cards to build an n-tier house of cards. - Martin Wohlgemuth, Aug 11 2002
The modular form Delta(q) = q*Product_{n>=1} (1-q^n)^24 = q*(1 + Sum_{n>=1} (-1)^n*(q^(n*(3*n-1)/2)+q^(n*(3*n+1)/2)))^24 = q*(1 + Sum_{n>=1} A033999(n)*(q^A000326(n)+q^a(n)))^24. - Jonathan Vos Post, Mar 15 2006
Row sums of triangle A134403.
Bisection of A001318. - Omar E. Pol, Aug 22 2011
Sequence found by reading the line from 0 in the direction 0, 7, ... and the line from 2 in the direction 2, 15, ... in the square spiral whose vertices are the generalized pentagonal numbers, A001318. - Omar E. Pol, Sep 08 2011
A general formula for the n-th second k-gonal number is given by T(n, k) = n*((k-2)*n+k-4)/2, n>=0, k>=5. - Omar E. Pol, Aug 04 2012
Partial sums give A006002. - Denis Borris, Jan 07 2013
A002260 is the following array A read by antidiagonals:
  0,  1,  2,  3,  4,  5, ...
  0,  1,  2,  3,  4,  5, ...
  0,  1,  2,  3,  4,  5, ...
  0,  1,  2,  3,  4,  5, ...
  0,  1,  2,  3,  4,  5, ...
  0,  1,  2,  3,  4,  5, ...
and a(n) is the hook sum: Sum_{k=0..n} A(n,k) + Sum_{r=0..n-1} A(r,n). - R. J. Mathar, Jun 30 2013
From Klaus Purath, May 13 2021: (Start)
This sequence and A000326 provide all integers m such that 24*m + 1 is a square. The union of the two sequences is A001318.
If A is a sequence satisfying the recurrence t(n) = 3*t(n-1) - 2*t(n-2) with the initial values either A(0) = 1, A(1) = n + 2 or A(0) = -1, A(1) = n - 1, then a(n) = (A(i)^2 - A(i-1)*A(i+1))/2^i + n^2 for i>0. (End)
a(n+1) is the number of Dyck paths of size (3,3n+2), i.e., the number of NE lattice paths from (0,0) to (3,3n+2) which stay above the line connecting these points. - Harry Richman, Jul 13 2021
Binomial transform of [0, 2, 3, 0, 0, 0, ...], being a(n) = 2*binomial(n,1) + 3*binomial(n,2). a(3) = 15 = [0, 2, 3, 0] dot [1, 3, 3, 1] = [0 + 6 + 9 + 0]. - Gary W. Adamson, Dec 17 2022
a(n) is the sum of longest side length of all nondegenerate integer-sided triangles with shortest side length n and middle side length (n + 1), n > 0. - Torlach Rush, Feb 04 2024

			From _Omar E. Pol_, Aug 22 2011: (Start)
Illustration of initial terms:
                                               O
                                             O O
                                 O         O O O
                               O O       O O O O
                     O       O O O     O O O O O
                   O O     O O O O     O O O O O
           O     O O O     O O O O     O O O O O
         O O     O O O     O O O O     O O O O O
    O    O O     O O O     O O O O     O O O O O
    O    O O     O O O     O O O O     O O O O O
    -    ---     -----     -------     ---------
    2     7        15         26           40
(End)

Henri Cohen, A Course in Computational Algebraic Number Theory, vol. 138 of Graduate Texts in Mathematics, Springer-Verlag, 2000.

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
A. O. L. Atkin and F. Morain, Elliptic Curves and Primality Proving, Math. Comp., Vol. 61, No. 203 (1993), pp. 29-68.
Charles H. Conley and Valentin Ovsienko, Quiddities of polygon dissections and the Conway-Coxeter frieze equation, arXiv:2107.01234 [math.CO], 2021.
Leonhard Euler, De mirabilibus proprietatibus numerorum pentagonalium, Acta Academiae Scientiarum Imperialis Petropolitanae, Vol. 1780: I, pp. 56-75.
Leonhard Euler, Observatio de summis divisorum, Novi Commentarii academiae scientiarum Petropolitanae, Vol. 5, pp. 59-74.
Leonhard Euler, An observation on the sums of divisors, arXiv:math/0411587 [math.HO], 2004-2009, p. 8.
Leonhard Euler, On the remarkable properties of the pentagonal numbers, arXiv:math/0505373 [math.HO], 2005.
Alfred Hoehn, Illustration of initial terms of A000326, A005449, A045943, A115067.
D. Suprijanto and I. W. Suwarno, Observation on Sums of Powers of Integers Divisible by 3k-1, Applied Mathematical Sciences, Vol. 8, No. 45 (2014), pp. 2211-2217.
Martin Wohlgemuth, Pentagon, Kartenhaus und Summenzerlegung.
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A016789 (first differences), A006002 (partial sums).
Cf. A000217, A000320, A000326, A001318, A033568, A049451, A101165, A101166.
The generalized pentagonal numbers b*n+3*n*(n-1)/2, for b = 1 through 12, form sequences A000326, this sequence, A045943, A115067, A140090, A140091, A059845, A140672-A140675, A151542.
Cf. numbers of the form n*(n*k-k+4)/2 listed in A226488 (this sequence is the case k=3).
Cf. numbers of the form n*((2*k+1)*n+1)/2 listed in A022289 (this sequence is the case k=1).

[n*(3*n + 1) / 2: n in [0..40]]; // Vincenzo Librandi, May 02 2011

A005449:=n->n*(3*n + 1)/2; seq(A005449(k), k=0..100); # Wesley Ivan Hurt, Oct 11 2013

Table[n (3 n + 1)/2, {n, 0, 100}] (* Zak Seidov, Jan 31 2012 *)

{a(n) = n * (3*n + 1) / 2} /* Michael Somos, Jul 18 2003 */

a(n) = A110449(n, 1) for n>0.
G.f.: x*(2+x)/(1-x)^3. E.g.f.: exp(x)*(2*x + 3*x^2/2). a(n) = n*(3*n + 1)/2. a(-n) = A000326(n). - Michael Somos, Jul 18 2003
a(n) = A001844(n) - A000217(n+1) = A101164(n+2,2) for n>0. - Reinhard Zumkeller, Dec 03 2004
a(n) = Sum_{j=1..n} n+j. - Zerinvary Lajos, Sep 12 2006
a(n) = A126890(n,n). - Reinhard Zumkeller, Dec 30 2006
a(n) = 2*C(3*n,4)/C(3*n,2), n>=1. - Zerinvary Lajos, Jan 02 2007
a(n) = A000217(n) + A000290(n). - Zak Seidov, Apr 06 2008
a(n) = a(n-1) + 3*n - 1 for n>0, a(0)=0. - Vincenzo Librandi, Nov 18 2010
a(n) = A129267(n+5,n). - Philippe Deléham, Dec 21 2011
a(n) = 2*A000217(n) + A000217(n-1). - Philippe Deléham, Mar 25 2013
a(n) = A130518(3*n+1). - Philippe Deléham, Mar 26 2013
a(n) = (12/(n+2)!)*Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*j^(n+2). - Vladimir Kruchinin, Jun 04 2013
a(n) = floor(n/(1-exp(-2/(3*n)))) for n>0. - Richard R. Forberg, Jun 22 2013
a(n) = Sum_{i=1..n} (3*i - 1) for n >= 1. - Wesley Ivan Hurt, Oct 11 2013 [Corrected by Rémi Guillaume, Oct 24 2024]
a(n) = (A000292(6*n+k+1)-A000292(k))/(6*n+1) - A000217(3*n+k+1), for any k >= 0. - Manfred Arens, Apr 26 2015
Sum_{n>=1} 1/a(n) = 6 - Pi/sqrt(3) - 3*log(3) = 0.89036376976145307522... . - Vaclav Kotesovec, Apr 27 2016
a(n) = A000217(2*n) - A000217(n). - Bruno Berselli, Sep 21 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*Pi/sqrt(3) + 4*log(2) - 6. - Amiram Eldar, Jan 18 2021
From Klaus Purath, May 13 2021: (Start)
Partial sums of A016789 for n > 0.
a(n) = 3*n^2 - A000326(n).
a(n) = A000326(n) + n.
a(n) = A002378(n) + A000217(n-1) for n >= 1. [Corrected by Rémi Guillaume, Aug 14 2024] (End)
From Klaus Purath, Jul 14 2021: (Start)
b^2 = 24*a(n) + 1 we get by b^2 = (a(n+1) - a(n-1))^2 = (a(2*n)/n)^2.
a(2*n) = n*(a(n+1) - a(n-1)), n > 0.
a(2*n+1) = n*(a(n+1) - a(n)). (End)
A generalization of Lajos' formula, dated Sep 12 2006, follows. Let SP(k,n) = the n-th second k-gonal number. Then SP(2k+1,n) = Sum_{j=1..n} (k-1)*n+j+k-2. - Charlie Marion, Jul 13 2024
a(n) = Sum_{k = 0..3*n} (-1)^(n+k+1) * binomial(k, 2) * binomial(3*n+k, 2*k). - Peter Bala, Nov 03 2024
For integer m, (6*m + 1)^2*a(n) + a(m) = a((6*m+1)*n + m). - Peter Bala, Jan 09 2025

A008288 Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 13, 7, 1, 1, 9, 25, 25, 9, 1, 1, 11, 41, 63, 41, 11, 1, 1, 13, 61, 129, 129, 61, 13, 1, 1, 15, 85, 231, 321, 231, 85, 15, 1, 1, 17, 113, 377, 681, 681, 377, 113, 17, 1, 1, 19, 145, 575, 1289, 1683, 1289, 575, 145, 19, 1, 1, 21, 181, 833, 2241, 3653, 3653

Offset: 0

N. J. A. Sloane

In the Formula section, some contributors use T(n,k) = D(n-k, k) (for 0 <= k <= n), which is the triangular version of the square array (D(n,k): n,k >= 0). Conversely, D(n,k) = T(n+k,k) for n,k >= 0. - Petros Hadjicostas, Aug 05 2020
Also called the tribonacci triangle [Alladi and Hoggatt (1977)]. - N. J. A. Sloane, Mar 23 2014
D(n,k) is the number of lattice paths from (0,0) to (n,k) using steps (1,0), (0,1), (1,1). - Joerg Arndt, Jul 01 2011 [Corrected by N. J. A. Sloane, May 30 2020]
Or, triangle read by rows of coefficients of polynomials P[n](x) defined by P[0] = 1, P[1] = x+1; for n >= 2, P[n] = (x+1)*P[n-1] + x*P[n-2].
D(n, k) is the number of k-matchings of a comb-like graph with n+k teeth. Example: D(1, 3) = 7 because the graph consisting of a horizontal path ABCD and the teeth Aa, Bb, Cc, Dd has seven 3-matchings: four triples of three teeth and the three triples {Aa, Bb, CD}, {Aa, Dd, BC}, {Cc, Dd, AB}. Also D(3, 1)=7, the 1-matchings of the same graph being the seven edges: {AB}, {BC}, {CD}, {Aa}, {Bb}, {Cc}, {Dd}. - Emeric Deutsch, Jul 01 2002
Sum of n-th antidiagonal of the array D is A000129(n+1). - Reinhard Zumkeller, Dec 03 2004 [Edited by Petros Hadjicostas, Aug 05 2020 so that the counting of antidiagonals of D starts at n = 0. That is, the sum of the terms in the n-th row of the triangles T is  A000129(n+1).]
The A-sequence for this Riordan type triangle (see one of Paul Barry's comments under Formula) is A112478 and the Z-sequence the trivial: {1, 0, 0, 0, ...}. See the W. Lang link under A006232 for Sheffer a- and z-sequences where also Riordan A- and Z-sequences are explained. This leads to the recurrence for the triangle given below. - Wolfdieter Lang, Jan 21 2008
The triangle or chess sums, see A180662 for their definitions, link the Delannoy numbers with twelve different sequences, see the crossrefs. All sums come in pairs due to the symmetrical nature of this triangle. The knight sums Kn14 and Kn15 have been added. It is remarkable that all knight sums are related to the tribonacci numbers, that is, A000073 and A001590, but none of the others. - Johannes W. Meijer, Sep 22 2010
This sequence, A008288, is jointly generated with A035607 as an array of coefficients of polynomials u(n,x): initially, u(1,x) = v(1,x) = 1; for n > 1, u(n,x) = x*u(n-1,x) + v(n-1) and v(n,x) = 2*x*u(n-1,x) + v(n-1,x). See the Mathematica section. - Clark Kimberling, Mar 09 2012
Row n, for n > 0, of Roger L. Bagula's triangle in the Example section shows the coefficients of the polynomial u(n) = c(0) + c(1)*x + ... + c(n)*x^n which is the numerator of the n-th convergent of the continued fraction [k, k, k, ...], where k = sqrt(x) + 1/sqrt(x); see A230000. - Clark Kimberling, Nov 13 2013
In an n-dimensional hypercube lattice, D(n,k) gives the number of nodes situated at a Minkowski (Manhattan) distance of k from a given node. In cellular automata theory, the cells at Manhattan distance k are called the von Neumann neighborhood of radius k. For k=1, see A005843. - Dmitry Zaitsev, Dec 10 2015
These numbers appear as the coefficients of series relating spherical and bispherical harmonics, in the solutions of Laplace's equation in 3D. [Majic 2019, Eq. 22] - Matt Majic, Nov 24 2019
From Peter Bala, Feb 19 2020: (Start)
The following remarks assume an offset of 1 in the row and column indices of the triangle.
The sequence of row polynomials T(n,x), beginning with T(1,x) = x, T(2,x) = x + x^2, T(3,x) = x + 3*x^2 + x^3, ..., is a strong divisibility sequence of polynomials in the ring Z[x]; that is, for all positive integers n and m, poly_gcd(T(n,x), T(m,x)) = T(gcd(n, m), x) - apply Norfleet (2005), Theorem 3. Consequently, the sequence (T(n,x): n >= 1) is a divisibility sequence in the polynomial ring Z[x]; that is, if n divides m then T(n,x) divides T(m,x) in Z[x].
Let S(x) = 1 + 2*x + 6*x^2 + 22*x^3 + ... denote the o.g.f. for the large Schröder numbers A006318. The power series (x*S(x))^n, n = 2, 3, 4, ..., can be expressed as a linear combination with polynomial coefficients of S(x) and 1: (x*S(x))^n = T(n-1,-x) - T(n,-x)*S(x). The result can be extended to negative integer n if we define T(0,x) = 0 and T(-n,x) = (-1)^(n+1) * T(n,x)/x^n. Cf. A115139.
[In the previous two paragraphs, D(n,x) was replaced with T(n,x) because the contributor is referring to the rows of the triangle T(n,k), not the rows of the array D(n,k). - Petros Hadjicostas, Aug 05 2020] (End)
Named after the French amateur mathematician Henri-Auguste Delannoy (1833-1915). - Amiram Eldar, Apr 15 2021
D(i,j) = D(j,i). With this and Dmitry Zaitsev's Dec 10 2015 comment, D(i,j) can be considered the number of points at L1 distance <= i in Z^j or the number of points at L1 distance <= j in Z^i from any given point. The rows and columns of D(i,j) are the crystal ball sequences on cubic lattices. See the first example below. The n-th term in the k-th crystal ball sequence can be considered the number of points at distance <= n from any point in a k-dimensional cubic lattice, or the number of points at distance <= k from any point in an n-dimensional cubic lattice. - Shel Kaphan, Jan 01 2023 and Jan 07 2023
Dimensions of hom spaces Hom(R^{(i)}, R^{(j)}) in the Delannoy category attached to the oligomorphic group of order preserving self-bijections of the real line. - Noah Snyder, Mar 22 2023

			The square array D(i,j) (i >= 0, j >= 0) begins:
  1, 1,  1,   1,   1,   1,    1,    1,    1,    1, ... = A000012
  1, 3,  5,   7,   9,  11,   13,   15,   17,   19, ... = A005408
  1, 5, 13,  25,  41,  61,   85,  113,  145,  181, ... = A001844
  1, 7, 25,  63, 129, 231,  377,  575,  833, 1159, ... = A001845
  1, 9, 41, 129, 321, 681, 1289, 2241, 3649, 5641, ... = A001846
  ...
For D(2,5) = 61, which is seen above in the row labeled A001844, we calculate the sum (9 + 11 + 41) of the 3 nearest terms above and/or to the left. - _Peter Munn_, Jan 01 2023
D(2,5) = 61 can also be obtained from the row labeled A005408 using a recurrence mentioned in the formula section:  D(2,5) = D(1,5) + 2*Sum_{k=0..4} D(1,k), so D(2,5) = 11 + 2*(1+3+5+7+9) = 11 + 2*25. - _Shel Kaphan_, Jan 01 2023
As a triangular array (on its side) this begins:
   0,   0,   0,   0,   1,   0,  11,   0, ...
   0,   0,   0,   1,   0,   9,   0,  61, ...
   0,   0,   1,   0,   7,   0,  41,   0, ...
   0,   1,   0,   5,   0,  25,   0, 129, ...
   1,   0,   3,   0,  13,   0,  63,   0, ...
   0,   1,   0,   5,   0,  25,   0, 129, ...
   0,   0,   1,   0,   7,   0,  41,   0, ...
   0,   0,   0,   1,   0,   9,   0,  61, ...
   0,   0,   0,   0,   1,   0,  11,   0, ...
   [Edited by _Shel Kaphan_, Jan 01 2023]
From _Roger L. Bagula_, Dec 09 2008: (Start)
As a triangle T(n,k) (with rows n >= 0 and columns k = 0..n), this begins:
   1;
   1,  1;
   1,  3,   1;
   1,  5,   5,   1;
   1,  7,  13,   7,    1;
   1,  9,  25,  25,    9,    1;
   1, 11,  41,  63,   41,   11,    1;
   1, 13,  61, 129,  129,   61,   13,   1;
   1, 15,  85, 231,  321,  231,   85,  15,   1;
   1, 17, 113, 377,  681,  681,  377, 113,  17,  1;
   1, 19, 145, 575, 1289, 1683, 1289, 575, 145, 19, 1;
   ... (End)
Triangle T(n,k) recurrence: 63 = T(6,3) = 25 + 13 + 25 = T(5,2) + T(4,2) + T(5,3).
Triangle T(n,k) recurrence with A-sequence A112478: 63 = T(6,3) = 1*25 + 2*25 - 2*9 + 6*1 (T entries from row n = 5 only). [Here the formula T(n,k) = Sum_{j=0..n-k} A112478(j) * T(n-1, k-1+j) is used with n = 6 and k = 3; i.e., T(6,3) = Sum_{j=0..3} A111478(j) * T(5, 2+j). - _Petros Hadjicostas_, Aug 05 2020]
From _Philippe Deléham_, Mar 29 2012: (Start)
Subtriangle of the triangle given by (1, 0, 1, -1, 0, 0, 0, ...) DELTA (0, 1, 0, 0, 0, ...) where DELTA is the operator defined in A084938:
   1;
   1,  0;
   1,  1,  0;
   1,  3,  1,  0;
   1,  5,  5,  1,  0;
   1,  7, 13,  7,  1,  0;
   1,  9, 25, 25,  9,  1, 0;
   1, 11, 41, 63, 41, 11, 1, 0;
   ...
Subtriangle of the triangle given by (0, 1, 0, 0, 0, ...) DELTA (1, 0, 1, -1, 0, 0, 0, ...) where DELTA is the operator defined in A084938:
   1;
   0, 1;
   0, 1,  1;
   0, 1,  3,  1;
   0, 1,  5,  5,  1;
   0, 1,  7, 13,  7,  1;
   0, 1,  9, 25, 25,  9,  1;
   0, 1, 11, 41, 63, 41, 11, 1;
   ... (End)

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Sums of antidiagonals: A000129 (Pell numbers).
Main diagonal: A001850 (central Delannoy numbers), which has further information and references.
A002002, A026002, and A190666 are +-k-diagonals for k=1, 2, 3 resp. - Shel Kaphan, Jan 01 2023
Rows 0..10: A000012, A005408, A001844, A001845, A001846, A001847, A001848, A001849, A008417, A008419, A008421.
See also A027618.
Cf. A059446.
Has same main diagonal as A064861. Different from A100936.
Cf. A101164, A101167, A128966, A131887, A131935.
Cf. A035607, A108625, A142979, A142992, A143007.
Read mod small primes: A211312, A211313, A211314, A211315.
Triangle sums (see the comments): A000129 (Row1); A056594 (Row2); A000073 (Kn11 & Kn21); A089068 (Kn12 & Kn22); A180668 (Kn13 & Kn23); A180669 (Kn14 & Kn24); A180670 (Kn15 & Kn25); A099463 (Kn3 & Kn4); A116404 (Fi1 & Fi2); A006498 (Ca1 & Ca2); A006498(3*n) (Ca3 & Ca4); A079972 (Gi1 & Gi2); A079972(4*n) (Gi3 & Gi4); A079973(3*n) (Ze1 & Ze2); A079973(2*n) (Ze3 & Ze4).
Cf. A102413, A128966. (D(n,1)) = A005843. Cf. A115139.

a008288 n k = a008288_tabl !! n !! k
a008288_row n = a008288_tabl !! n
a008288_tabl = map fst $ iterate
    (\(us, vs) -> (vs, zipWith (+) ([0] ++ us ++ [0]) $
                       zipWith (+) ([0] ++ vs) (vs ++ [0]))) ([1], [1, 1])
-- Reinhard Zumkeller, Jul 21 2013

A008288 := proc(n, k) option remember; if k = 0 then 1 elif n=k then 1 else procname(n-1, k-1) + procname(n-2, k-1) + procname(n-1, k) end if; end proc: seq(seq(A008288(n,k),k=0..n), n=0..10); # triangular indices n and k
P[0]:=1; P[1]:=x+1; for n from 2 to 12 do P[n]:=expand((x+1)*P[n-1]+x*P[n-2]); lprint(P[n]); lprint(seriestolist(series(P[n],x,200))); end do:

(* Next, A008288 jointly generated with A035607 *)
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];
v[n_, x_] := 2 x*u[n - 1, x] + v[n - 1, x];
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A008288 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A035607 *)
(* Clark Kimberling, Mar 09 2012 *)
d[n_, k_] := Binomial[n+k, k]*Hypergeometric2F1[-k, -n, -n-k, -1]; A008288 = Flatten[Table[d[n-k, k], {n, 0, 12}, {k, 0, n}]] (* Jean-François Alcover, Apr 05 2012, after 3rd formula *)

from functools import cache
@cache
def delannoy_row(n: int) -> list[int]:
    if n == 0: return [1]
    if n == 1: return [1, 1]
    rov = delannoy_row(n - 2)
    row = delannoy_row(n - 1) + [1]
    for k in range(n - 1, 0, -1):
        row[k] += row[k - 1] + rov[k - 1]
    return row
for n in range(10): print(delannoy_row(n))  # Peter Luschny, Jul 30 2023

for k in range(8):  # seen as an array, read row by row
    a = lambda n: hypergeometric([-n, -k], [1], 2)
    print([simplify(a(n)) for n in range(11)]) # Peter Luschny, Nov 19 2014

D(n, 0) = 1 = D(0, n) for n >= 0; D(n, k) = D(n, k-1) + D(n-1, k-1) + D(n-1, k).
Bivariate o.g.f.: Sum_{n >= 0, k >= 0} D(n, k)*x^n*y^k = 1/(1 - x - y - x*y).
D(n, k) = Sum_{d = 0..min(n,k)} binomial(k, d)*binomial(n+k-d, k) = Sum_{d=0..min(n,k)} 2^d*binomial(n, d)*binomial(k, d). [Edited by Petros Hadjicostas, Aug 05 2020]
Seen as a triangle read by rows: T(n, 0) = T(n, n) = 1 for n >= 0 and T(n, k) = T(n-1, k-1) + T(n-2, k-1) + T(n-1, k), 0 < k < n and n > 1. - Reinhard Zumkeller, Dec 03 2004
Read as a number triangle, this is the Riordan array (1/(1-x), x(1+x)/(1-x)) with T(n, k) = Sum_{j=0..n-k} C(n-k, j) * C(k, j) * 2^j. - Paul Barry, Jul 18 2005
T(n,k) = Sum_{j=0..n-k} C(k,j)*C(n-j,k). - Paul Barry, May 21 2006
Let y^k(n) be the number of Khalimsky-continuous functions f from [0,n-1] to Z such that f(0) = 0 and f(n-1) = k. Then y^k(n) = D(i,j) for i = (1/2)*(n-1-k) and j = (1/2)*(n-1+k) where n-1+k belongs to 2Z. - Shiva Samieinia (shiva(AT)math.su.se), Oct 08 2007
Recurrence for triangle from A-sequence (see the Wolfdieter Lang comment above): T(n,k) = Sum_{j=0..n-k} A112478(j) * T(n-1, k-1+j), n >= 1, k >= 1. [For k > n, the sum is empty, in which case T(n,k) = 0.]
From Peter Bala, Jul 17 2008: (Start)
The n-th row of the square array is the crystal ball sequence for the product lattice A_1 x ... x A_1 (n copies). A035607 is the table of the associated coordination sequences for these lattices.
The polynomial p_n(x) := Sum {k = 0..n} 2^k * C(n,k) * C(x,k) = Sum_{k = 0..n} C(n,k) * C(x+k,n), whose values [p_n(0), p_n(1), p_n(2), ... ] give the n-th row of the square array, is the Ehrhart polynomial of the n-dimensional cross polytope (the hyperoctahedron) [Bump et al. (2000), Theorem 6].
The first few values are p_0(x) = 1, p_1(x) = 2*x + 1, p_2(x) = 2*x^2 + 2*x + 1 and p_3(x) = (4*x^3 + 6*x^2 + 8*x + 3)/3.
The reciprocity law p_n(m) = p_m(n) reflects the symmetry of the table.
The polynomial p_n(x) is the unique polynomial solution of the difference equation (x+1)*f(x+1) - x*f(x-1) = (2*n+1)*f(x), normalized so that f(0) = 1.
These polynomials have their zeros on the vertical line Re x = -1/2 in the complex plane; that is, the polynomials p_n(x-1), n = 1,2,3,..., satisfy a Riemann hypothesis [Bump et al. (2000), Theorem 4]. The o.g.f. for the p_n(x) is (1 + t)^x/(1 - t)^(x + 1) = 1 + (2*x + 1)*t + (2*x^2 + 2*x + 1)*t^2 + ... .
The square array of Delannoy numbers has a close connection with the constant log(2). The entries in the n-th row of the array occur in the series acceleration formula log(2) = (1 - 1/2 + 1/3 - ... + (-1)^(n+1)/n) + (-1)^n * Sum_{k>=1} (-1)^(k+1)/(k*D(n,k-1)*D(n,k)). [T(n,k) was replaced with D(n,k) in the formula to agree with the beginning of the paragraph. - Petros Hadjicostas, Aug 05 2020]
For example, the fourth row of the table (n = 3) gives the series log(2) = 1 - 1/2 + 1/3 - 1/(1*1*7) + 1/(2*7*25) - 1/(3*25*63) + 1/(4*63*129) - ... . See A142979 for further details.
Also the main diagonal entries (the central Delannoy numbers) give the series acceleration formula Sum_{n>=1} 1/(n*D(n-1,n-1)*D(n,n)) = (1/2)*log(2), a result due to Burnside. [T(n,n) was replaced here with D(n,n) to agree with the previous paragraphs. - Petros Hadjicostas, Aug 05 2020]
Similar relations hold between log(2) and the crystal ball sequences of the C_n lattices A142992. For corresponding results for the constants zeta(2) and zeta(3), involving the crystal ball sequences for root lattices of type A_n and A_n x A_n, see A108625 and A143007 respectively. (End)
From Peter Bala, Oct 28 2008: (Start)
Hilbert transform of Pascal's triangle A007318 (see A145905 for the definition of this term).
D(n+a,n) = P_n(a,0;3) for all integer a such that a >= -n, where P_n(a,0;x) is the Jacobi polynomial with parameters (a,0) [Hetyei]. The related formula A(n,k) = P_k(0,n-k;3) defines the table of asymmetric Delannoy numbers, essentially A049600. (End)
Seen as a triangle read by rows: T(n, k) = Hyper2F1([k-n, -k], [1], 2). - Peter Luschny, Aug 02 2014, Oct 13 2024.
From Peter Bala, Jun 25 2015: (Start)
O.g.f. for triangle T(n,k): A(z,t) = 1/(1 - (1 + t)*z - t*z^2) = 1 + (1 + t)*z + (1 + 3*t + t^2)*z^2 + (1 + 5*t + 5*t^2 + t^3)*z^3 + ....
1 + z*d/dz(A(z,t))/A(z,t) is the o.g.f. for A102413. (End)
E.g.f. for the n-th subdiagonal of T(n,k), n >= 0, equals exp(x)*P(n,x), where P(n,x) is the polynomial Sum_{k = 0..n} binomial(n,k)*(2*x)^k/k!. For example, the e.g.f. for the second subdiagonal is exp(x)*(1 + 4*x + 4*x^2/2) = 1 + 5*x + 13*x^2/2! + 25*x^3/3! + 41*x^4/4! + 61*x^5/5! + .... - Peter Bala, Mar 05 2017 [The n-th subdiagonal of triangle T(n,k) is the n-th row of array D(n,k).]
Let a_i(n) be multiplicative with a_i(p^e) = D(i, e), p prime and e >= 0, then Sum_{n > 0} a_i(n)/n^s = (zeta(s))^(2*i+1)/(zeta(2*s))^i for i >= 0. - Werner Schulte, Feb 14 2018
Seen as a triangle read by rows: T(n,k) = Sum_{i=0..k} binomial(n-i, i) * binomial(n-2*i, k-i) for 0 <= k <= n. - Werner Schulte, Jan 09 2019
Univariate generating function: Sum_{k >= 0} D(n,k)*z^k = (1 + z)^n/(1 - z)^(n+1). [Dziemianczuk (2013), Eq. 5.3] - Matt Majic, Nov 24 2019
(n+1)*D(n+1,k) = (2*k+1)*D(n,k) + n*D(n-1,k). [Majic (2019), Eq. 22] - Matt Majic, Nov 24 2019
For i, j >= 1, D(i,j) = D(i,j-1) + 2*Sum_{k=0..i-1} D(k,j-1), or, because D(i,j) = D(j,i), D(i,j) = D(i-1,j) + 2*Sum_{k=0..j-1} D(i-1,k). - Shel Kaphan, Jan 01 2023
Sum_{k=0..n} T(n,k)^2 = A026933(n). - R. J. Mathar, Nov 07 2023
Let S(x) = (1 - x - (1 - 6*x + x^2)^(1/2))/(2*x) denote the g.f. of the sequence of large Schröder numbers A006318. Read as a lower triangular array, the signed n-th row polynomial R(n, -x) = 1/sqrt(1 - 6*x + x^2) *( 1/S(x)^(n+1) + (x*S(x))^(n+1) ). For example, R(4, -x) = 1 - 7*x + 13*x^2 - 7*x^3 + x^4 = 1/sqrt(1 - 6*x + x^2) * ( 1/S(x)^5 + (x*S(x))^5 ). Cf. A102413. - Peter Bala, Aug 01 2024

Expanded description from Clark Kimberling, Jun 15 1997
Additional references from Sylviane R. Schwer (schwer(AT)lipn.univ-paris13.fr), Nov 28 2001
Changed the notation to make the formulas more precise. - N. J. A. Sloane, Jul 01 2002

A008574 a(0) = 1, thereafter a(n) = 4n.

Original entry on oeis.org

1, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 128, 132, 136, 140, 144, 148, 152, 156, 160, 164, 168, 172, 176, 180, 184, 188, 192, 196, 200, 204, 208, 212, 216, 220, 224, 228, 232

Offset: 0

N. J. A. Sloane; entry revised Aug 24 2014

Number of squares on the perimeter of an (n+1) X (n+1) board. - Jon Perry, Jul 27 2003
Coordination sequence for square lattice (or equivalently the planar net 4.4.4.4).
Apparently also the coordination sequence for the planar net 3.4.6.4. - Darrah Chavey, Nov 23 2014
From N. J. A. Sloane, Nov 26 2014: (Start)
I confirm that this is indeed the coordination sequence for the planar net 3.4.6.4. The points at graph distance n from a fixed point in this net essentially lie on a hexagon (see illustration in link).
If n = 3k, k >= 1, there are 2k + 1 nodes on each edge of the hexagon. This counts the corners of the hexagon twice, so the number of points in the shell is 6(2k + 1) - 6 = 4n. If n = 3k + 1, the numbers of points on the six edges of the hexagon are 2k + 2 (4 times) and 2k + 1 (twice), for a total of 12k + 10 - 6 = 4n. If n = 3k + 2 the numbers are 2k + 2 (4 times) and 2k + 3 twice, and again we get 4n points.
The illustration shows shells 0 through 12, as well as the hexagons formed by shells 9 (green, 36 points), 10 (black, 40 points), 11 (red, 44 points), and 12 (blue, 48 points).
It is clear from the net that this period-3 structure continues forever, and establishes the theorem.
In contrast, for the 4.4.4.4 planar net, the successive shells are diamonds instead of hexagons, and again the n-th shell (n > 0) contains 4n points.
Of course the two nets are very different, since 4.4.4.4 has the symmetry of the square, while 3.4.6.4 has only mirror symmetry (with respect to a point), and has the symmetry of a regular hexagon with respect to the center of any of the 12-gons. (End)
Also the coordination sequence for a 6.6.6.6 point in the 3-transitive tiling {4.6.6, 6.6.6, 6.6.6.6}, see A265045, A265046. - N. J. A. Sloane, Dec 27 2015
Also the coordination sequence for 2-dimensional cyclotomic lattice Z[zeta_4].
Susceptibility series H_1 for 2-dimensional Ising model (divided by 2).
Also the Engel expansion of exp^(1/4); cf. A006784 for the Engel expansion definition. - Benoit Cloitre, Mar 03 2002
This sequence differs from A008586, multiples of 4, only in its initial term. - Alonso del Arte, Apr 14 2011
Number of 2 X n binary matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00,0), (00;1) and (10;1). An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1 < i2 and j1 < j2 and these elements are in same relative order as those in the triple (x,y,z). - Sergey Kitaev, Nov 11 2004
Central terms of the triangle in A118013. - Reinhard Zumkeller, Apr 10 2006
Also the coordination sequence for the htb net. - N. J. A. Sloane, Mar 31 2018
This is almost certainly also the coordination sequence for Dual(3.3.4.3.4) with respect to a tetravalent node. - Tom Karzes, Apr 01 2020
Minimal number of segments (equivalently, corners) in a rook circuit of a 2n X 2n board (maximal number is A085622). - Ruediger Jehn, Jan 02 2021

			From _Omar E. Pol_, Aug 20 2011 (Start):
Illustration of initial terms as perimeters of squares (cf. Perry's comment above):
.                                         o o o o o o
.                             o o o o o   o         o
.                   o o o o   o       o   o         o
.           o o o   o     o   o       o   o         o
.     o o   o   o   o     o   o       o   o         o
. o   o o   o o o   o o o o   o o o o o   o o o o o o
.
. 1    4      8        12         16           20
(End)

T. D. Noe, Table of n, a(n) for n = 0..1000
Joerg Arndt, The 3.4.6.4 net
Matthias Beck and Serkan Hosten, Cyclotomic polytopes and growth series of cyclotomic lattices, arXiv:math/0508136 [math.CO], 2005-2006.
Pierre de la Harpe, On the prehistory of growth of groups, arXiv:2106.02499 [math.GR], 2021.
Jean-Guillaume Eon, Symmetry and Topology: The 11 Uninodal Planar Nets Revisited, Symmetry, 10 (2018), 13 pages, doi:10.3390/sym10020035. See Section 7.
Brian Galebach, k-uniform tilings (k <= 6) and their A-numbers
Chaim Goodman-Strauss and N. J. A. Sloane, A Coloring Book Approach to Finding Coordination Sequences, Acta Cryst. A75 (2019), 121-134, also on NJAS's home page. Also on arXiv, arXiv:1803.08530 [math.CO], 2018-2019.
Rostislav Grigorchuk and Cosmas Kravaris, On the growth of the wallpaper groups, arXiv:2012.13661 [math.GR], 2020. See section 4.2 p. 20.
Branko Grünbaum and Geoffrey C. Shephard, Tilings by regular polygons, Mathematics Magazine, 50 (1977), 227-247.
A. J. Guttmann, Indicators of solvability for lattice models, Discrete Math., 217 (2000), 167-189.
D. Hansel et al., Analytical properties of the anisotropic cubic Ising model, J. Stat. Phys., 48 (1987), 69-80.
Tom Karzes, Tiling Coordination Sequences
Sergey Kitaev, On multi-avoidance of right angled numbered polyomino patterns, Integers: Electronic Journal of Combinatorial Number Theory 4 (2004), A21, 20pp.
Reticular Chemistry Structure Resource, sql and htb
Anton Shutov and Andrey Maleev, Coordination sequences of 2-uniform graphs, Z. Kristallogr., 235 (2020), 157-166. See supplementary material, krb, vertex u_1.
N. J. A. Sloane, Illustration of points in shells 0 through 12 of the 3.4.6.4 planar net (see Comments for discussion)
N. J. A. Sloane, The uniform planar nets and their A-numbers [Annotated scanned figure from Gruenbaum and Shephard (1977)]
N. J. A. Sloane, Overview of coordination sequences of Laves tilings [Fig. 2.7.1 of Grünbaum-Shephard 1987 with A-numbers added and in some cases the name in the RCSR database]
N. J. A. Sloane, Coordination Sequences, Planing Numbers, and Other Recent Sequences (II), Experimental Mathematics Seminar, Rutgers University, Jan 31 2019, Part I, Part 2, Slides. (Mentions this sequence)
Michael Somos, Rational Function Multiplicative Coefficients
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A001844 (partial sums), A008586, A054275, A054410, A054389, A054764.
Convolution square of A040000.
Row sums of A130323 and A131032.
List of coordination sequences for uniform planar nets: A008458 (the planar net 3.3.3.3.3.3), A008486 (6^3), A008574 (4.4.4.4 and 3.4.6.4), A008576 (4.8.8), A008579(3.6.3.6), A008706 (3.3.3.4.4), A072154 (4.6.12), A219529(3.3.4.3.4), A250120 (3.3.3.3.6), A250122 (3.12.12).
List of coordination sequences for Laves tilings (or duals of uniform planar nets): [3,3,3,3,3.3] = A008486; [3.3.3.3.6] = A298014, A298015, A298016; [3.3.3.4.4] = A298022, A298024; [3.3.4.3.4] = A008574, A296368; [3.6.3.6] = A298026, A298028; [3.4.6.4] = A298029, A298031, A298033; [3.12.12] = A019557, A298035; [4.4.4.4] = A008574; [4.6.12] = A298036, A298038, A298040; [4.8.8] = A022144, A234275; [6.6.6] = A008458.
Coordination sequences for the 20 2-uniform tilings in the order in which they appear in the Galebach catalog, together with their names in the RCSR database (two sequences per tiling): #1 krt A265035, A265036; #2 cph A301287, A301289; #3 krm A301291, A301293; #4 krl A301298, A298024; #5 krq A301299, A301301; #6 krs A301674, A301676; #7 krr A301670, A301672; #8 krk A301291, A301293; #9 krn A301678, A301680; #10 krg A301682, A301684; #11 bew A008574, A296910; #12 krh A301686, A301688; #13 krf A301690, A301692; #14 krd A301694, A219529; #15 krc A301708, A301710; #16 usm A301712, A301714; #17 krj A219529, A301697; #18 kre A301716, A301718; #19 krb A301720, A301722; #20 kra A301724, A301726.
See also A265045, A265046.

a008574 0 = 1; a008574 n = 4 * n
a008574_list = 1 : [4, 8 ..]  -- Reinhard Zumkeller, Apr 16 2015

f[0] = 1; f[n_] := 4 n; Array[f, 59, 0] (* or *)
CoefficientList[ Series[(1 + x)^2/(1 - x)^2, {x, 0, 58}], x] (* Robert G. Wilson v, Jan 02 2011 *)
Join[{1},Range[4,232,4]] (* Harvey P. Dale, Aug 19 2011 *)
a[ n_] := 4 n + Boole[n == 0]; (* Michael Somos, Jan 07 2019 *)

{a(n) = 4*n + !n}; /* Michael Somos, Apr 16 2007 */

Binomial transform is A000337 (dropping the 0 there). - Paul Barry, Jul 21 2003
Euler transform of length 2 sequence [4, -2]. - Michael Somos, Apr 16 2007
G.f.: ((1 + x) / (1 - x))^2. E.g.f.: 1 + 4*x*exp(x). - Michael Somos, Apr 16 2007
a(-n) = -a(n) unless n = 0. - Michael Somos, Apr 16 2007
G.f.: exp(4*atanh(x)). - Jaume Oliver Lafont, Oct 20 2009
a(n) = a(n-1) + 4, n > 1. - Vincenzo Librandi, Dec 31 2010
a(n) = A005408(n-1) + A005408(n), n > 1. - Ivan N. Ianakiev, Jul 16 2012
a(n) = 4*n = A008586(n), n >= 1. - Tom Karzes, Apr 01 2020

A005900 Octahedral numbers: a(n) = n(2n^2 + 1)/3.

Original entry on oeis.org

0, 1, 6, 19, 44, 85, 146, 231, 344, 489, 670, 891, 1156, 1469, 1834, 2255, 2736, 3281, 3894, 4579, 5340, 6181, 7106, 8119, 9224, 10425, 11726, 13131, 14644, 16269, 18010, 19871, 21856, 23969, 26214, 28595, 31116, 33781, 36594, 39559, 42680

Offset: 0

N. J. A. Sloane

Series reversion of g.f.: A(x) is Sum_{n>0} - A066357(n)(-x)^n.
Partial sums of centered square numbers A001844. - Paul Barry, Jun 26 2003
Also as a(n) = (1/6)*(4n^3 + 2n), n>0: structured tetragonal diamond numbers (vertex structure 5) (cf. A000447 - structured diamonds); and structured trigonal anti-prism numbers (vertex structure 5) (cf. A100185 - structured anti-prisms). Cf. A100145 for more on structured polyhedral numbers. - James A. Record (james.record(AT)gmail.com), Nov 07 2004
Schlaefli symbol for this polyhedron: {3,4}.
If X is an n-set and Y and Z are disjoint 2-subsets of X then a(n-4) is equal to the number of 5-subsets of X intersecting both Y and Z. - Milan Janjic, Aug 26 2007
Starting with 1 = binomial transform of [1, 5, 8, 4, 0, 0, 0, ...] where (1, 5, 8, 4) = row 3 of the Chebyshev triangle A081277. - Gary W. Adamson, Jul 19 2008
a(n) = largest coefficient of (1 + ... + x^(n-1))^4. - R. H. Hardin, Jul 23 2009
Convolution square root of (1 + 6x + 19x^3 + ...) = (1 + 3x + 5x^2 + 7x^3 + ...) = A005408(x). - Gary W. Adamson, Jul 27 2009
Starting with offset 1 = the triangular series convolved with [1, 3, 4, 4, 4, ...]. - Gary W. Adamson, Jul 28 2009
One of the 5 Platonic polyhedral (tetrahedral, cube, octahedral, dodecahedral, and icosahedral) numbers (cf. A053012). - Daniel Forgues, May 14 2010
Let b be any product of four different primes. Then the divisor lattice of b^n is of width a(n+1). - Jean Drabbe, Oct 13 2010
Arises in Bezdek's proof on contact numbers for congruent sphere packings (see preprint). - Jonathan Vos Post, Feb 08 2011
Euler transform of length 2 sequence [6, -2]. - Michael Somos, Mar 27 2011
a(n+1) is the number of 2 X 2 matrices with all terms in {0,1,...,n} and (sum of terms) = 2n. - Clark Kimberling, Mar 19 2012
a(n) is the number of semistandard Young tableaux over all partitions of 3 with maximal element <= n. - Alois P. Heinz, Mar 22 2012
Self convolution of the odd numbers. - Reinhard Zumkeller, Apr 04 2012
a(n) is the number of (w,x,y,z) with all terms in {1,...,n} and w+x=y+z; also the number of (w,x,y,z) with all terms in {0,...,n} and |w-x|<=y. - Clark Kimberling, Jun 02 2012
The sequence is the third partial sum of (0, 1, 3, 4, 4, 4, ...). - Gary W. Adamson, Sep 11 2015
a(n) is the number of join-irreducible elements in the Weyl group of type B_n with respect to the strong Bruhat order. - Rafael Mrden, Aug 26 2020
Number of unit octahedra contained in an n-scale octahedron composed of a tetrahedral-octahedral honeycomb. The number of unit tetrahedra in it is 8*A000292(n-1) = 4*(n^3 - n)/3. Also, the number of unit tetrahedra and unit octahedra contained in an n-scale tetrahedron composed of a tetrahedral-octahedral honeycomb is respectively A006527(n) = (n^3 + 2*n)/3 and A000292(n-1) = (n^3 - n)/6. - Jianing Song, Feb 24 2025

			G.f. = x + 6*x^2 + 19*x^3 + 44*x^4 + 85*x^5 + 146*x^6 + 231*x^7 + ...

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 50.
H. S. M. Coxeter, Polyhedral numbers, pp. 25-35 of R. S. Cohen, J. J. Stachel and M. W. Wartofsky, eds., For Dirk Struik: Scientific, historical and political essays in honor of Dirk J. Struik, Reidel, Dordrecht, 1974.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
X. Acloque, Polynexus Numbers and other mathematical wonders [broken link]
Karoly Bezdek, Contact numbers for congruent sphere packings, arXiv:1102.1198 [math.MG], 2011.
Matteo Cavaleri and Alfredo Donno, Some degree and distance-based invariants of wreath products of graphs, arXiv:1805.08989 [math.CO], 2018.
Y-h. Guo, Some n-Color Compositions, J. Int. Seq. 15 (2012) 12.1.2, eq (5), m=2.
Milan Janjic, Two Enumerative Functions
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
Muhammad Fatih Killik and Bünyamin Şahi̇n, Further Results on Level Matrix, Int'l J. Math. Combin. (2024) Vol 4, 68-74. See p. 71.
Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2002), 65-75.
Hankyung Ko, Volodymyr Mazorchuk and Rafael Mrđen, Join operation for the Bruhat order and Verma modules, arXiv:2109.01067 [math.RT], 2021. See Remark 5.10 p. 19.
A. Lascoux and M.-P. Schützenberger, Treillis et bases des groupes de Coxeter, Electron. J. Combin. 3 (1996), #R27.
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (11).
J. K. Merikoski, R. Kumar and R. A. Rajput, Upper bounds for the largest eigenvalue of a bipartite graph, Electronic Journal of Linear Algebra ISSN 1081-3810, A publication of the International Linear Algebra Society, Volume 26, pp. 168-176, April 2013.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.
Eric Weisstein's World of Mathematics, Octahedral Number.
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Sums of 2 consecutive terms give A001845. Cf. A001844.
1/12*t*(n^3-n)+n for t = 2, 4, 6, ... gives A004006, A006527, A006003, A005900, A004068, A000578, A004126, A000447, A004188, A004466, A004467, A007588, A062025, A063521, A063522, A063523.
Cf. A022521.
Cf. A081277.
Row n=3 of A210391. - Alois P. Heinz, Mar 22 2012
Cf. A005408.
Cf. A053676, A053677, A053678.
Cf. A002061.
Cf. A000292 (tetrahedral numbers), A000578 (cubes), A006566 (dodecahedral numbers), A006564 (icosahedral numbers).
Similar sequence: A014820(n-1) (m=4), A069038 (m=5), A069039 (m=6), A099193(m=7), A099195 (m=8), A099196 (m=9), A099197 (m=10).

a005900 n = sum $ zipWith (*) odds $ reverse odds
            where odds = take n a005408_list
a005900_list = scanl (+) 0 a001844_list
-- Reinhard Zumkeller, Jun 16 2013, Apr 04 2012

[n*(2*n^2+1)/3: n in [0..50]]; // Wesley Ivan Hurt, Sep 11 2015

I:=[0,1,6,19]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Sep 12 2015

al:=proc(s,n) binomial(n+s-1,s); end; be:=proc(d,n) local r; add( (-1)^r*binomial(d-1,r)*2^(d-1-r)*al(d-r,n), r=0..d-1); end; [seq(be(3,n), n=0..100)];
A005900:=(z+1)**2/(z-1)**4; # Simon Plouffe in his 1992 dissertation
with(combinat): seq(fibonacci(4,2*n)/12, n=0..40); # Zerinvary Lajos, Apr 21 2008

Table[(2n^3+n)/3, {n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1}, {0,1,6,19},50] (* Harvey P. Dale, Oct 10 2013 *)
CoefficientList[Series[x (1 + x)^2/(1 - x)^4, {x, 0, 45}], x] (* Vincenzo Librandi, Sep 12 2015 *)

makelist(n*(2*n^2+1)/3, n, 0, 20); /* Martin Ettl, Jan 07 2013 */

PARI
```
{a(n) = n*(2*n^2+1)/3};
```

concat([0],Vec(x*(1 + x)^2/(1 - x)^4 + O(x^50))) \\ Indranil Ghosh, Mar 16 2017

def a(n): return n*(2*n*n + 1)//3
print([a(n) for n in range(41)]) # Michael S. Branicky, Sep 03 2021

a(n) = 1^2 + 2^2 + ... + (n-1)^2 + n^2 + (n-1)^2 + ... + 2^2 + 1^2. - Amarnath Murthy, May 28 2001
G.f.: x * (1 + x)^2 / (1 - x)^4. a(n) = -a(-n) = (2*n^3 + n) / 3.
a(n) = ( ((n+1)^5-n^5) - (n^5-(n-1)^5) )/30. - Xavier Acloque, Oct 17 2003
a(n) is the sum of the products pq, where p and q are both positive and odd and p + q = 2n, e.g., a(4) = 7*1 + 5*3 + 3*5 + 1*7 = 44. - Jon Perry, May 17 2005
a(n) = 4*binomial(n,3) + 4*binomial(n,2) + binomial(n,1). - Mitch Harris, Jul 06 2006
a(n) = binomial(n+2,3) + 2*binomial(n+1,3) + binomial(n,3), (this pair generalizes; see A014820, the 4-cross polytope numbers).
Sum_{n>=1} 1/a(n) = 3*gamma + 3*Psi((I*(1/2))*sqrt(2)) - (1/2)*(3*I)*Pi*coth((1/2)*Pi*sqrt(2)) - (1/2)*(3*I)*sqrt(2) = A175577, where I=sqrt(-1). - Stephen Crowley, Jul 14 2009
a(n) = A035597(n)/2. - J. M. Bergot, Jun 11 2012
a(n) = A000578(n) - 2*A000292(n-1) for n>0. - J. M. Bergot, Apr 05 2014
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), n>3. - Wesley Ivan Hurt, Sep 11 2015
E.g.f.: (1/3)*x*(3 + 6*x + 2*x^2)*exp(x). - Ilya Gutkovskiy, Mar 16 2017
a(n) = (A002061(A002061(n+1)) - A002061(A002061(n)))/6. - Daniel Poveda Parrilla, Jun 10 2017
a(n) = 6*a(n-1)/(n-1) + a(n-2) for n > 1. - Seiichi Manyama, Jun 06 2018
Sum_{n >= 1} (-1)^(n+1)/(a(n)*a(n+1)) = 6*log(2) - 4 = 1/(6 + 2/(6 + 6/(6 + ... + n*(n-1)/(6 + ...)))). See A142983. - Peter Bala, Mar 06 2024

A000982 a(n) = ceiling(n^2/2).

Original entry on oeis.org

0, 1, 2, 5, 8, 13, 18, 25, 32, 41, 50, 61, 72, 85, 98, 113, 128, 145, 162, 181, 200, 221, 242, 265, 288, 313, 338, 365, 392, 421, 450, 481, 512, 545, 578, 613, 648, 685, 722, 761, 800, 841, 882, 925, 968, 1013, 1058, 1105, 1152, 1201, 1250, 1301, 1352, 1405

Offset: 0

N. J. A. Sloane

a(n) = number of pairs (i,j) in [1..n] X [1..n] with integral arithmetic mean. Cf. A132188, A362931. - N. J. A. Sloane, Aug 28 2023
Also, floor( (n^2+1)/2 ). - N. J. A. Sloane, Feb 08 2019
Floor(arithmetic mean of next n numbers). - Amarnath Murthy, Mar 11 2003
Pairwise sums of repeated squares (A008794).
Also, number of topologies on n+1 unlabeled elements with exactly 4 elements in the topology. a(3) gives 4 elements a,b,c,d; the valid topologies are (0,a,ab,abcd), (0,a,abc,abcd), (0,ab,abc,abcd), (0,a,bcd,abcd) and (0,ab,cd,abcd), with a count of 5. - Jon Perry, Mar 05 2004
Partition n into two parts, say, r and s, so that r^2 + s^2 is minimal, then a(n) = r^2 + s^2. Geometrical significance: folding a rod with length n units at right angles in such a way that the end points are at the least distance, which is given by a(n)^(1/2) as the hypotenuse of a right triangle with the sum of the base and height = n units. - Amarnath Murthy, Apr 18 2004
Convolution of A002061(n)-0^n and (-1)^n. Convolution of n (A001477) with {1,0,2,0,2,0,2,...}. Partial sums of repeated odd numbers {0,1,1,3,3,5,5,...}. - Paul Barry, Jul 22 2004
The ratio of the sum of terms over the total number of terms in an n X n spiral. The sum of terms of an n X n spiral is A037270, or Sum_{k=0..n^2} k = (n^4 + n^2)/2 and the total number of terms is n^2. - William A. Tedeschi, Feb 27 2008
Starting with offset 1 = row sums of triangle A158946. - Gary W. Adamson, Mar 31 2009
Partial sums of A109613. - Reinhard Zumkeller, Dec 05 2009
Also the number of compositions of even natural numbers into 2 parts < n. For example a(3)=5 are the compositions (0,0), (0,2), (2,0), (1,1), (2,2) of even natural numbers into 2 parts < 3. a(4)=8 are the compositions (0,0), (0,2), (2,0), (1,1), (2,2), (1,3), (3,1), (3,3) of even natural numbers into 2 parts < 4. - Adi Dani, Jun 05 2011
A001105 and A001844 interleaved. - Omar E. Pol, Sep 18 2011
Number of (w,x,y) having all terms in {0,...,n} and w=average(x,y). - Clark Kimberling, May 15 2012
For n > 0, minimum number of lines necessary to get through all unit cubes of an n X n X n cube (see Kantor link). - Michel Marcus, Apr 13 2013
Sum_{n > 0} 1/a(n) = Sum_{n > 0} 1/(2*n^2) + Sum_{n >= 0} 1/(2*n + 2*n^2 + 1) = (zeta(2) + (Pi* tanh(Pi/2)))/2 = 2.26312655.... - Enrique Pérez Herrero, Jun 17 2013
For n > 1, a(n) is the edge cover number of the n X n king graph. - Eric W. Weisstein, Jun 20 2017
Also the number of vertices in the n X n black bishop graph. - Eric W. Weisstein, Jun 26 2017
The same sequence arises in the triangular array of the integers >= 1, according to a simple "zig-zag" rule for selection of terms. a(n-1) lies in the (n-1)-th row of the array, and the second row of that sub-array (with apex a(n-1)) contains just two numbers, one odd, one even. The one with opposite parity to a(n-1) is a(n). - David James Sycamore, Jul 29 2018
Size of minimal ternary 1-covering code with code length n, i.e., K_n(3,1). See Kalbfleisch and Stanton. - Patrick Wienhöft, Jan 29 2019
For n > 1, a(n-1) is the maximum number of inversions in a permutation consisting of a single n-cycle on n symbols. - M. Ryan Julian Jr., Sep 10 2019
Also the number of classes of convex inscribed polyominoes in a (2,n) rectangular grid; two polyominoes are in the same class if one of them can be obtained by a reflection or 180-degree rotation of the other. - Jean-Luc Manguin, Jan 29 2020
a(n) is the number of pairs (p,q) such that 1 <= p, p+1 < q <= n+2 and q <> 2*p. - César Eliud Lozada, Oct 25 2020
a(n) is the maximum number of copies of a 12 permutation pattern in an alternating (or zig-zag) permutation of length n+1. The maximum number of copies of 123 in an alternating permutation is motivated in the Notices reference, and the argument here is analogous. - Lara Pudwell, Dec 01 2020
It appears that a(n) is the largest number of nodes of an induced path in the n X n king graph. An induced path going in a simple spiraling pattern, starting in a corner, has a(n) nodes. For even n this is optimal, because an induced path can have at most two nodes in any 2 X 2 subsquare. For odd n, I cannot see how to prove that (n^2+1)/2 is best possible. See also A357501. - Pontus von Brömssen, Oct 02 2022 [Proved by Beluhov (2023). - Pontus von Brömssen, Jan 30 2023]
a(n) = n + 2*(n-2) + 2*(n-4) + 2*(n-6) + ... number of black squares on an n X n chessboard. - R. J. Mathar, Dec 03 2022

			G.f. = x + 2*x^2 + 5*x^3 + 8*x^4 + 13*x^5 + 18*x^6 + 25*x^7 + 32*x^8 + ...
Centrosymmetric 3 X 3 matrix: [[a,b,c],[d,e,d],[c,b,a]], a(3) = 3*(3-1)/2 + (3-1)/2 + 1 = (3^2+1)/2 = 5 from a,b,c,d,e. 4 X 4 case: [[a,b,c,d],[e,f,g,h],[h,g,f,e],[d,c,b,a]], a(4) = 4*4/2 = 8. - _Wolfdieter Lang_, Oct 12 2015
a(3) = 5. The alternating permutation of length 3 + 1 = 4 with the maximum number of copies of 123 is 1324. The five copies are 12, 13, 14, 23, and 24. - _Lara Pudwell_, Dec 01 2020

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..3000
Nikolai Beluhov, Snake paths in king and knight graphs, arXiv:2301.01152 [math.CO], 2023.
M. Benoumhani and M. Kolli, Finite topologies and partitions, JIS 13 (2010) # 10.3.5, t_{N0}(n,4) in theorem 5.
Andrea C. Burgess, Caleb W. Jones, and David A. Pike, Extending Graph Burning to Hypergraphs, arXiv:2403.01001 [math.CO], 2024. See p. 9.
Geoffrey B. Campbell, Vector Partition Identities for 2D, 3D and nD Lattices, arXiv:2302.01091 [math.CO], 2023.
Ronald Cools, Ian H. Sloan, Minimial cubature formulae of trigonometric degree, Math. Comp. 65 (216) (1996) 1583-1600. Table 1 dimension 2.
John Elias, Illustration of Initial Terms: Intersection of a double spaced square grid and centrally aligned triangle.
J. G. Kalbfleisch and R. G. Stanton, A combinatorial problem in matching, J. London Math. Soc. Vol. 1, No. 1 (1969), 60-64. [Corrected by _N. J. A. Sloane_, Feb 08 2019]
J. M. Kantor, Mathématiques venues d'ailleurs: divertissements mathématiques en U.R.S.S., Le cube transpercé, pp. 56-62, Belin, Paris, 1982.
S. Lafortune, A. Ramani, B. Grammaticos, Y. Ohta, and K.M. Tamizhmani, Blending two discrete integrability criteria: ..., arXiv:nlin/0104020 [nlin.SI], 2001.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Lara Pudwell, From permutation patterns to the periodic table, Notices of the American Mathematical Society, 67.7 (2020), 994-1001.
Eric Weisstein's World of Mathematics, Black Bishop Graph
Eric Weisstein's World of Mathematics, Edge Cover Number
Eric Weisstein's World of Mathematics, King Graph
Eric Weisstein's World of Mathematics, Topology
Eric Weisstein's World of Mathematics, Vertex Count
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A000096, A000217, A001105, A001477, A001844, A002061, A004526, A005843, A007590, A008794, A037270, A081352, A109613, A110654, A116940, A134444, A158946, A168380, A357501.
Column 2 of A195040.
Cf. also A132188, A362931.

a000982 = (`div` 2) . (+ 1) . (^ 2)  -- Reinhard Zumkeller, Jun 27 2013

[(2*n^2 + 1 - (-1)^n) / 4: n in [0..60]]; // Vincenzo Librandi, Jun 16 2011

seq( ceil(n^2/2),n=0..30) ; # R. J. Mathar, Jun 05 2011

Table[Ceiling[n^2/2], {n, 0, 120}] (* Vladimir Joseph Stephan Orlovsky, Apr 02 2011 *)
Accumulate[Join[{0}, (# - Boole[EvenQ[#]] &) /@ Range[80]]] (* Alonso del Arte, Sep 11 2019 *)

a(n)=(n^2+1)\2 \\ Charles R Greathouse IV, Sep 13 2013

x='x+O('x^100); concat([0], Vec(x*(1+x^2)/((1+x)*(1-x)^3))) \\ Altug Alkan, Oct 12 2015

apply( A000982(n)=n^2\/2, [0..55]) \\ M. F. Hasler, Feb 29 2020

def A000982(n): return n**2+1>>1 # Chai Wah Wu, Aug 28 2023

(((1 to 49) by 2) flatMap { List.fill(2)() }).scanLeft(0)( + ) // _Alonso del Arte, Sep 11 2019

a(2*n) = 2*n^2, a(2*n+1) = 2*n^2 + 2*n + 1.
G.f.: -x*(1+x^2) / ( (1+x)*(x-1)^3 ). - Simon Plouffe in his 1992 dissertation
From Benoit Cloitre, Nov 06 2002: (Start)
a(n) = (2*n^2 + 1 - (-1)^n) / 4.
a(0)=0, a(1)=1; for n>1, a(n+1) = n + 1 + max(2*floor(a(n)/2), 3*floor(a(n)/3)). (End)
G.f.: (x + x^2 + x^3 + x^4)/((1 - x)*(1 - x^2)^2), not reduced. - Len Smiley
a(n) = a(n-2) + 2n - 2. - Paul Barry, Jul 17 2004
From Paul Barry, Jul 22 2004: (Start)
G.f.: x*(1+x^2)/((1-x^2)*(1-x)^2) = x*(1+x^2)/((1+x)*(1-x)^3);
a(n) = Sum_{k=0..n} (k^2 - k + 1 - 0^k)*(-1)^(n-k);
a(n) = Sum_{k=0..n} (1 + (-1)^(n-k) - 0^(n-k))*k. (End)
From Reinhard Zumkeller, Feb 27 2006: (Start)
a(0) = 0, a(n+1) = a(n) + 2*floor(n/2) + 1.
a(n) = A116940(n) - A005843(n). (End)
Starting with offset 1, = row sums of triangle A134444. Also, with offset 1, = binomial transform of [1, 1, 2, -2, 4, -8, 16, -32, ...]. - Gary W. Adamson, Oct 25 2007
a(n) = floor((n^2+1)/2). - William A. Tedeschi, Feb 27 2008
a(n) = A004526(n+1) + A000217(n-1). - Yosu Yurramendi, Sep 12 2008, corrected by Klaus Purath, Jun 15 2021
From Jaume Oliver Lafont, Dec 05 2008: (Start)
a(n) = a(n-1) + a(n-2) - a(n-3) + 2.
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4). (End)
a(n) = A004526(n)^2 + A110654(n)^2. - Philippe Deléham, Mar 12 2009
a(n) = n^2 - floor(n^2/2). - Wesley Ivan Hurt, Jun 14 2013
Euler transform is length 4 sequence [2, 2, 0, -1].
a(n) = a(-n) for all n in Z. - Michael Somos, May 05 2015
a(n) is also the number of independent entries in a centrosymmetric n X n matrix: M(i, j) = M(n-i+1, n-j+1). - Wolfdieter Lang, Oct 12 2015
For n > 1, a(n+1)/a(n) = 3 - A081352(n-2)/a(n). - Miko Labalan, Mar 26 2016
E.g.f.: (1/2)*(x*(1 + x)*cosh(x) + (1 + x + x^2)*sinh(x)). - Stefano Spezia, Feb 03 2020
a(n) = binomial(n+1,2) - floor(n/2). - César Eliud Lozada, Oct 25 2020
From Klaus Purath, Jun 15 2021: (Start)
a(n-1) + a(n) = A002061(n).
a(n) = (a(n-1)^2 + 1) / a(n-2), n >= 3 odd.
a(n) = (a(n-1)^2 - (n-1)^2) / a(n-2), n >= 4 even. (End)

A001788 a(n) = n(n+1)2^(n-2).

Original entry on oeis.org

0, 1, 6, 24, 80, 240, 672, 1792, 4608, 11520, 28160, 67584, 159744, 372736, 860160, 1966080, 4456448, 10027008, 22413312, 49807360, 110100480, 242221056, 530579456, 1157627904, 2516582400, 5452595200, 11777605632, 25367150592, 54492397568, 116769423360, 249644974080, 532575944704

Offset: 0

N. J. A. Sloane

Number of 2-dimensional faces in (n+1)-dimensional hypercube; also number of 4-cycles in the (n+1)-dimensional hypercube. - Henry Bottomley, Apr 14 2000
Also the number of edges in the (n+1)-halved cube graph. - Eric W. Weisstein, Jun 21 2017
From Philippe Deléham, Apr 28 2004: a(n) is the sum, over all nonempty subsets E of {1, 2, ..., n}, of all elements of E. E.g., a(3) = 24: the nonempty subsets are {1, 2, 3}, {1, 2}, {1, 3}, {2, 3}, {1}, {2}, {3} and 1 + 2 + 3 + 1 + 2 + 1 + 3 + 2 + 3 + 1 + 2 + 3 = 24.
Equivalently, sum of all nodes (except the last one, equal to n+1) of all integer compositions of n+1. - Olivier Gérard, Oct 22 2011
The inverse binomial transform of a(n-k) for k=-1..4 gives A001844, A000290, A000217(n-1), A002620(n-1), A008805(n-4), A000217 interspersed with 0's. - Michael Somos, Jul 18 2003
Take n points on a finite line. They all move with the same constant speed; they instantaneously change direction when they collide with another; and they fall when they quit the line. a(n-1) is the total number of collisions before falling when the initials directions are the 2^n possible. The mean number of collisions is then n(n-1)/8. E.g., a(1)=0 before any collision is possible. a(2)=1 because there is a collision only if the initials directions are, say, right-left. - Emmanuel Moreau, Feb 11 2006
Also number of pericondensed hexagonal systems with n hexagons. For example, if n=5 then the number of pericondensed hexagonal systems with n hexagons is 24. - Parthasarathy Nambi, Sep 06 2006
If X_1,X_2,...,X_n is a partition of a 2n-set X into 2-blocks then, for n>1, a(n-1) is equal to the number of (n+2)-subsets of X intersecting each X_i (i=1,2,...,n). - Milan Janjic, Jul 21 2007
Number of n-permutations of 3 objects u,v,w, with repetition allowed, containing exactly two u's. Example: a(2)=6 because we have uuw, uuv, uwu, uvu, wuu and vuu. - Zerinvary Lajos, Dec 29 2007
For n>0 where [0]={}, the empty set, and [n]={1,2,...n} a(n) is the number of ways to separate [n-1] into three non-overlapping intervals (allowed to be empty) and then choose a subset from each interval. - Geoffrey Critzer, Feb 07 2009
Form an array with m(n,0) = m(0,n) = n^2 and m(i,j) = m(i-1,j-1) + m(i-1,j). Then m(1,n) = A001844(n) and m(n,n) = a(n). - J. M. Bergot, Nov 07 2012
The sum of the number of inversions of all sequences of zeros and ones with length n+1. - Evan M. Bailey, Dec 09 2020
a(n) is the number of strings of length n defined on {0,1,2,3} that contain at most one 2, exactly one 3, and have no restriction on the number of 0s and 1s. For example, a(3)=24 since the strings are 321 (6 of this type), 320 (6 of this type), 310 (6 of this type), 300 (3 of this type) and 311 (3 of this type). - Enrique Navarrete, May 04 2025

			The nodes of an integer composition are the partial sums of its elements, seen as relative distances between nodes of a 1-dimensional polygon. For a composition of 7 such as 1+2+1+3, the nodes are 0,1,3,4,7. Their sum (without the last node) is 8. The sum of all nodes of all 2^(7-1)=64 integer compositions of 7 is 672.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 796.
Clifford A. Pickover, The Math Book, From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics, Sterling Publ., NY, 2009, page 282.
A. P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..500
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
H. J. Brothers, Pascal's Prism: Supplementary Material.
Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
Robert Davis and Greg Simay, Further Combinatorics and Applications of Two-Toned Tilings, arXiv:2001.11089 [math.CO], 2020.
Herbert Izbicki, Über Unterbäume eines Baumes, Monatshefte fur Mathematik, Vol. 74 (1970), pp. 56-62.
Milan Janjic, Two Enumerative Functions.
Milan Janjic and Boris Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
C. W. Jones, J. C. P. Miller, J. F. C. Conn, and R. C. Pankhurst, Tables of Chebyshev polynomials, Proc. Roy. Soc. Edinburgh. Sect. A., Vol. 62, No. 2 (1946), pp. 187-203.
Han Mao Kiah, Alexander Vardy, and Hanwen Yao, Efficient Algorithms for the Bee-Identification Problem, arXiv:2212.09952 [cs.IT], 2022.
Duško Letić, Nenad Cakić, Branko Davidović, Ivana Berković and Eleonora Desnica, Some certain properties of the generalized hypercubical functions, Advances in Difference Equations, Vol. 2011 (2011), Article 60.
Mircea Merca, A Special Case of the Generalized Girard-Waring Formula, J. Integer Sequences, Vol. 15 (2012), Article 12.5.7.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Lara Pudwell, Nathan Chenette and Manda Riehl, Statistics on Hypercube Orientations, AMS Special Session on Experimental and Computer Assisted Mathematics, Joint Mathematics Meetings (Denver 2020).
John Riordan and N. J. A. Sloane, Correspondence, 1974.
R. Tosic, D. Masulovic, I. Stojmenovic, J. Brunvoll, B. N. Cyvin and S. J. Cyvin, Enumeration of polyhex hydrocarbons to h = 17, J. Chem. Inf. Comput. Sci., Vol. 35, No. 2 (1995), pp. 181-187.
Eric Weisstein's World of Mathematics, Edge Count.
Eric Weisstein's World of Mathematics, Graph Cycle.
Eric Weisstein's World of Mathematics, Idempotent Number.
Eric Weisstein's World of Mathematics, Halved Cube Graph.
Eric Weisstein's World of Mathematics, Hypercube Graph.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Cf. A000079, A001787, A001789, A001793 (sum of all nodes of integer compositions, n included).
Cf. A001844, A038207, A290031 (6-cycles).
Row sums of triangle A094305.
Sequences similar to the form q^(n-2)*binomial(n, 2): A000217 (q=1), this sequence (q=2), A027472 (q=3), A038845 (q=4), A081135 (q=5), A081136 (q=6), A027474 (q=7), A081138 (q=8), A081139 (q=9), A081140 (q=10), A081141 (q=11), A081142 (q=12), A027476 (q=15).

List([0..30], n-> n*(n+1)*2^(n-2)); # G. C. Greubel, Aug 27 2019

a001788 n = if n < 2 then n else n * (n + 1) * 2 ^ (n - 2)
a001788_list = zipWith (*) a000217_list $ 1 : a000079_list
-- Reinhard Zumkeller, Jul 11 2014

[n*(n+1)*2^(n-2): n in [0..30]]; // G. C. Greubel, Aug 27 2019

A001788 := n->n*(n+1)*2^(n-2);
A001788:=-1/(2*z-1)**3; # Simon Plouffe in his 1992 dissertation; gives sequence without initial zero

CoefficientList[Series[x/(1-2x)^3, {x,0,30}], x]
Table[n*(n+1)*2^(n-2), {n,0,30}]
With[{n = 30}, Join[{0}, Times @@@ Thread[{Accumulate[Range[n]], 2^Range[0, n - 1]}]]] (* Harvey P. Dale, Jul 16 2013 *)
LinearRecurrence[{6, -12, 8}, {0, 1, 6}, 30] (* Harvey P. Dale, Jul 16 2013 *)

PARI
```
a(n)=if(n<0,0,2^n*n*(n+1)/4)
```

A001788_upto(n)=Vec(x/(1-2*x)^3+O(x^n),-n) \\ for illustration. - M. F. Hasler, Oct 05 2024

[n if n < 2 else n * (n + 1) * 2**(n - 2) for n in range(28)] # Zerinvary Lajos, Mar 10 2009

G.f.: x/(1-2*x)^3.
E.g.f.: x*(1 + x)*exp(2*x).
a(n) = 2*a(n-1) + n*2^(n-1) = 2*a(n-1) + A001787(n).
a(n) = A038207(n+1,2).
a(n) = A055252(n, 2).
a(n) = Sum_{i=1..n} i^2 * binomial(n, i): binomial transform of A000290. - Yong Kong, Dec 26 2000
a(n) = Sum_{j=0..n} binomial(n+1,j)*(n+1-j)^2. - Zerinvary Lajos, Aug 22 2006
If the leading 0 is deleted, the binomial transform of A001844: (1, 5, 13, 25, 41, ...); = double binomial transform of [1, 4, 4, 0, 0, 0, ...]. - Gary W. Adamson, Sep 02 2007
a(n) = Sum_{1<=i<=k<=n} (-1)^(i+1)*i^2*binomial(n+1,k+i)*binomial(n+1,k-i). - Mircea Merca, Apr 09 2012
a(0)=0, a(1)=1, a(2)=6, a(n) = 6*a(n-1) - 12*a(n-2) + 8*a(n-3). - Harvey P. Dale, Jul 16 2013
a(n) = Sum_{k=0..n-1} Sum_{i=0..n-1} (k+1) * C(n-1,i). - Wesley Ivan Hurt, Sep 20 2017
From Amiram Eldar, Jan 05 2022: (Start)
Sum_{n>=1} 1/a(n) = 4*(1-log(2)).
Sum_{n>=1} (-1)^(n+1)/a(n) = 12*log(3/2) - 4. (End)

cp's OEIS Frontend

A001653 Numbers k such that 2*k^2 - 1 is a square.

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A008586 Multiples of 4.

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A046092 4 times triangular numbers: a(n) = 2*n*(n+1).

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A005448 Centered triangular numbers: a(n) = 3*n*(n-1)/2 + 1.

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A005449 Second pentagonal numbers: a(n) = n*(3*n + 1)/2.

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A046092 4 times triangular numbers: a(n) = 2n(n+1).

A005448 Centered triangular numbers: a(n) = 3n(n-1)/2 + 1.

A005449 Second pentagonal numbers: a(n) = n(3n + 1)/2.

A005900 Octahedral numbers: a(n) = n(2n^2 + 1)/3.