A005448 -id:A005448 - cp's OEIS frontend

A077043 "Three-quarter squares": a(n) = n^2 - A002620(n).

0, 1, 3, 7, 12, 19, 27, 37, 48, 61, 75, 91, 108, 127, 147, 169, 192, 217, 243, 271, 300, 331, 363, 397, 432, 469, 507, 547, 588, 631, 675, 721, 768, 817, 867, 919, 972, 1027, 1083, 1141, 1200, 1261, 1323, 1387, 1452, 1519, 1587, 1657, 1728, 1801, 1875, 1951

Offset: 0

Henry Bottomley, Oct 22 2002

Triangular numbers plus quarter squares: (n+1)*(n+2)/2 + floor(n^2/4) (i.e., A000217(n+1) + A002620(n)).
Largest coefficient in the expansion of (1+x+x^2+...+x^(n-1))^3=((1-x^n)/(1-x))^3, i.e., the coefficient of x^floor[3(n-1)/2] and of x^ceiling[3(n-1)/2]; also number of compositions of [3(n+1)/2] into exactly 3 positive integers each no more than n.
A set of n independent statements a,b,c,d..., produces n^2 conditional statements of the form "If a, then b" (including self-implications such as "If a, then a"). If such statements are taken as equivalent to "It is not the case that the first statement is true and the second is false" (material implication), A077043(n) is the minimum number of the conditional statements that can be true. (The maximum number of false conditional statements is A002620(n), the maximum product of two integers whose sum is n.) - Matthew Vandermast, Mar 04 2003
This is also the maximum number of triple intersections between three sets of n lines, where the lines in each set are parallel to each other. E.g., for n=3:
\.\.\.../././
.\.\.\./././.
..\.\.x././..
---+-*-*-+---
----*-*-*----
---+-*-*-+---
.././.x.\.\..
./././.\.\.\.
/././...\.\.\
where '*' = triple intersection, '+' and 'x' = double intersection.
I am pretty sure that the hexagonal configuration of intersections shown above is the optimum and I get the formulas a(n) = (3n^2)/4 for n even and (3n^2+1)/4 for n odd. - Gabriel Nivasch (gnivasch(AT)yahoo.com), Jan 13 2004
For n > 1 the sequence represents the maximum number of points that can be placed in a plane such that the largest distances between any two points does not exceed the shortest of the distances between any two points by more than a factor n-1. - Johannes Koelman (Joc_kay(AT)hotmail.com), Apr 27 2006
This is also the number of distinct noncongruent isosceles triangles with side length up to n. - Patrick Hurst (patrick(AT)imsa.edu), May 14 2008
Also concentric triangular numbers. A033428 and A003215 interleaved. - Omar E. Pol, Sep 28 2011
Number of (w,x,y) with all terms in {0,...,n} and w=x>range{w,x,y}. - Clark Kimberling, Jun 02 2012
Number of pairs (x,y) with x in {0,...,n}, y even in {0,...,2n}, and x<=y. - Clark Kimberling, Jul 02 2012
From Bob Selcoe, Aug 05 2013: (Start)
a(n) is the number of 3-member sets with non-repeating positive integer values (x,y,z) whose sums equal 3(n+1). Example: a(4)=12; thus there are 12 sets where x+y+z = 15: (1,2,12), (1,3,11), (1,4,10), (1,5,9), (1,6,8), (2,3,10), (2,4,9), (2,5,8), (2,6,7), (3,4,8), (3,5,7) and (4,5,6).
From above, the number of sets sharing minimum values (minvals) equals a(1)-a(0), a(2)-a(1), a(3)-a(2),... a(n)-a(n-1) which are the numbers not divisible by 3, in sequence (A001651), range n to 1. So in the above example, there is one set with minval 4, two sets with minval 3, four sets with minval 2 and five sets with minval 1. (End)
Number of partitions of 3n into exactly 3 parts. - Wesley Ivan Hurt, Jan 21 2014
Number of partitions of 3(n-1) into at most 3 parts. - Colin Barker, Mar 31 2015
Number of possible positions after n-1 steps on the lines of a hexagonal grid. - Reg Robson, Mar 08 2014
12*a(n) is a perfect square when n is even and 12*a(n) - 3 is a perfect square when n is odd. - Miquel Cerda, Jun 30 2016
Square of largest Euclidean distance from start point reachable by an n-step walk on a honeycomb lattice. - Hugo Pfoertner, Jun 21 2018

			G.f. = x + 3*x^2 + 7*x^3 + 12*x^4 + 19*x^5 + 27*x^6 + 37*x^7 + 48*x^8 + ...
a(4)=12 since the compositions of floor(3*(4+1)/2) = 7 into exactly 3 positive integers each no more than 4 are 1+2+4, 1+3+3, 1+4+2, 2+1+4, 2+2+3, 2+3+3, 2+4+1, 3+1+3, 3+2+2, 3+3+1, 4+1+2, 4+2+1.
From _Philippe Deléham_, Dec 17 2011: (Start)
a(1) = 1 = 1^3;
a(1) + a(3) = 1 + 7 = 2^3;
a(1) + a(3) + a(5) = 1 + 7 + 19 = 3^3;
a(1) + a(3) + a(5) + a(7) = 1 + 7 + 19 + 37 = 4^3;
a(1) + a(3) + a(5) + a(7) + a(9) = 1 + 7 + 19 + 37 + 61 = 5^3; ... (End)

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 64.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Ralf Hinze, Concrete stream calculus: An extended study, J. Funct. Progr. 20 (5-6) (2010) 463-535, doi, Section 4.4.4.
Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7 (2004), Article 04.1.6.
Gabriel Nivasch and Eyal Lev, Nonattacking Queens on a Triangle, Mathematics Magazine, Vol. 78, No. 5 (Dec., 2005), pp. 399-403. See page 402.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).
Index entries for sequences related to compositions.

Column 3 of A195040. - Omar E. Pol, Sep 28 2011
Cf. A019298 (partial sums).
Cf. A002620, A033428, A003215.
Equals one more than A331952 and one less than A084684. - Greg Dresden, Feb 22 2020

a077043 n = a077043_list !! n
a077043_list = scanl (+) 0 a001651_list
-- Reinhard Zumkeller, Jan 06 2014

[Ceiling(n^2*3/4): n in [0..60]]; // Vincenzo Librandi, Jun 29 2011

A077043:=n->ceil(3*n^2/4); seq(A077043(n), n=0..60); # Wesley Ivan Hurt, Jan 21 2014

Table[Ceiling[(3n^2)/4], {n,0,60}] (* or *) LinearRecurrence[{2,0,-2,1}, {0,1,3,7}, 60] (* Harvey P. Dale, Dec 16 2012 *)

{a(n) = n^2 - (n^2 \ 4)}; /* Michael Somos, Jun 29 2011 */

a(n) = ceiling(n^2*3/4) = A077042(n, 3); a(-n) = a(n).
Also can be computed from 1 * C(n,0) + 2 * C(n,1) + 2 * C(n,2) - Sum((-2)^(k-3) C(n, k)). - Joshua Zucker, Nov 10 2002
a(n) = A002620(n-1) + A002620(n) + A002620(n+1). - Jon Perry, May 29 2003
From Jon Perry, May 29 2003: (Start)
a(2k)   = a(2k-2) + 6k - 3,
a(2k+1) = a(2k-1) + 6k,
a(4n)   = 12n^2,
a(4n+1) = a(4n)   + 6n + 1,
a(4n+2) = a(4n+1) + 6n + 2,
a(4n+3) = a(4n+2) + 6n + 4,
a(4n+4) = a(4n+3) + 6n + 5.
Differences between alternate terms give 3, 6, 9, 12, ... (End)
a(n+1) - a(n) = A001651(n), partial sums of A001651. - Reinhard Zumkeller, Dec 28 2007
From R. J. Mathar, Nov 10 2008: (Start)
G.f.: x*(1+x+x^2)/((1+x)*(1-x)^3).
a(n) + a(n+1) = A005448(n+1).
The inverse binomial transform yields 0 followed by A141531. (End)
Euler transform of length 3 sequence [3, 1, -1]. - Michael Somos, Jun 29 2011
a(n) = 3*n^2/4 - ((-1)^n-1)/8. - Omar E. Pol, Sep 28 2011
Sum_{k=0..n} a(2k+1) = partial sums of A003215 = (n+1)^3 (see example). - Philippe Deléham, Dec 17 2011
a(0)=0, a(1)=1, a(2)=3, a(3)=7, a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4). - Harvey P. Dale, Dec 16 2012
a(0)=0, a(1)=1, a(n) = 3*(n-1) + a(n-2). - Reg Robson, Mar 08 2014
a(2k) = 3k^2 = A033428(k), a(2k+1) = 3k^2 + 3k + 1 = A003215(k). - Jon Perry, Oct 25 2014
a(n) = Sum_{j=1..n} Sum_{i=1..n} ceiling((i+j-n)/2). - Wesley Ivan Hurt, Mar 12 2015
a(n) = (3*n)^2/12 for n even and a(n) = ((3*n)^2 + 3)/12 for n odd. - Miquel Cerda, Jun 30 2016
a(n) = Sum_{k=1..n} floor((n+k)/2). - Wesley Ivan Hurt, Mar 31 2017
0 = 1 +a(n)*(+a(n+1) -a(n+2)) +a(n+1)*(-3 -a(n+1) +a(n+2)) for all n in Z. - Michael Somos, Apr 02 2017
E.g.f.: (1/8)*exp(-x)*(-1 + exp(2*x)*(1 + 6*x + 6*x^2)). - Stefano Spezia, Nov 29 2019
Sum_{n>=1} 1/a(n) = Pi^2/18 + tanh(Pi/(2*sqrt(3)))*Pi/sqrt(3). - Amiram Eldar, Jan 16 2023

A068601 a(n) = n^3 - 1.

Original entry on oeis.org

0, 7, 26, 63, 124, 215, 342, 511, 728, 999, 1330, 1727, 2196, 2743, 3374, 4095, 4912, 5831, 6858, 7999, 9260, 10647, 12166, 13823, 15624, 17575, 19682, 21951, 24388, 26999, 29790, 32767, 35936, 39303, 42874, 46655, 50652, 54871, 59318, 63999, 68920

Offset: 1

Naohiro Nomoto, Mar 28 2002

a(n) is the least positive integer k such that k can only contain 'n-1' in exactly 2 different bases B, where 1 < B <= k.
Apart from the first term, the same as A135300. - R. J. Mathar, Apr 29 2008
A058895(n)^3 + a(n)^3 + A033562(n)^3 = A185065(n)^3. - Vincenzo Librandi, Mar 13 2012
Numbers k such that for every nonnegative integer m, k^(3*m+1) + k^(3*m) is a cube. - Arkadiusz Wesolowski, Aug 10 2013

			For n=6; 215 written in bases 6 and 42 is 555, 55 and (555, 55) are exactly 2 different bases.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000217, A003215, A005448, A016921, A033562, A058895, A129294, A135300, A185065, A339604.

List([1..45],n->n^3-1); # Muniru A Asiru, Oct 23 2018

[n^3-1: n in [1..40]]; // Vincenzo Librandi, Mar 11 2012

A068601:=n->n^3-1: seq(A068601(n), n=1..50); # Wesley Ivan Hurt, Jul 23 2014

f[n_]:=n^3-1;f[Range[60]] (* Vladimir Joseph Stephan Orlovsky, Feb 14 2011*)
LinearRecurrence[{4,-6,4,-1},{0,7,26,63},50] (* Vincenzo Librandi, Mar 11 2012 *)
Range[50]^3 - 1 (* Wesley Ivan Hurt, Jul 23 2014 *)

PARI
```
a(n)=n^3-1
```

for n in range(1,50): print(n**3-1, end=', ') # Stefano Spezia, Nov 21 2018

Partial sums of A003215, hex (or centered hexagonal) numbers: 3*n(n+1)+1. - Jonathan Vos Post, Mar 16 2006
G.f.: x^2*(7-2*x+x^2)/(1-x)^4. - Colin Barker, Feb 12 2012
4*a(m^2-2*m+2) = (m^2-m+1)^3 + (m^2-m-1)^3 + (m^2-3*m+3)^3 + (m^2-3*m+1)^3. - Bruno Berselli, Jun 23 2014
a(n) = Sum_{i=1..n-1} (i+1)^3 - i^3. - Wesley Ivan Hurt, Jul 23 2014
Sum_{n>=2} 1/a(n) = Sum_{n>=1} (zeta(3*n) - 1) = A339604. - Amiram Eldar, Nov 06 2020
Product_{n>=2} (1 + 1/a(n)) = 3*Pi*sech(sqrt(3)*Pi/2). - Amiram Eldar, Jan 20 2021
E.g.f.: 1 + exp(x)*(x^3 + 3*x^2 + x - 1). - Stefano Spezia, Jul 06 2021

A069129 Centered 16-gonal numbers.

Original entry on oeis.org

1, 17, 49, 97, 161, 241, 337, 449, 577, 721, 881, 1057, 1249, 1457, 1681, 1921, 2177, 2449, 2737, 3041, 3361, 3697, 4049, 4417, 4801, 5201, 5617, 6049, 6497, 6961, 7441, 7937, 8449, 8977, 9521, 10081, 10657, 11249, 11857, 12481, 13121, 13777, 14449, 15137, 15841

Offset: 1

Terrel Trotter, Jr., Apr 07 2002

Also, sequence found by reading the line from 1, in the direction 1, 17, ..., in the square spiral whose vertices are the triangular numbers A000217. Opposite numbers to the members of A139098 in the same spiral. - Omar E. Pol, Apr 26 2008
The subsequence of primes begins: 17, 97, 241, 337, 449, 577, 881, 1249, 3041, 3361, 3697, 4049, 4801, 6961, 7937, 9521,  10657, 13121, 14449. See A184899: n such that the n-th centered 12-gonal number is prime. Indices of prime star numbers. - Jonathan Vos Post, Feb 27 2011
Binomial transform of [1, 16, 16, 0, 0, 0, ...] and Narayana transform (A001263) of [1, 16, 0, 0, 0, ...]. - Gary W. Adamson, Jul 28 2011
Centered hexadecagonal numbers or centered hexakaidecagonal numbers. - Omar E. Pol, Oct 03 2011
a(n) = m(n,n) for an array constructed by using the terms in A016813 as the antidiagonals; the first few antidiagonals are 1; 5,9; 13,17,21; 25,29,33,37. - J. M. Bergot, Jul 05 2013
[The first five rows begin: 1,9,21,37,57; 5,17,33,53,77; 13,29,49,73,101; 25,45,69,97,129; 41,65,93,125,161.]

			a(5) = 161 because 8*5^2 - 8*5 + 1 = 200 - 40 + 1 = 161.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.
Leo Tavares, Illustration: Octagonal Stars.
Eric Weisstein's World of Mathematics, Centered Polygonal Numbers.
R. Yin, J. Mu, and T. Komatsu, The p-Frobenius Number for the Triple of the Generalized Star Numbers, Preprints 2024, 2024072280. See p. 2.
Index entries for sequences related to centered polygonal numbers.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005448, A001844, A005891, A003215, A069099, A000217, A035008, A139098.

Bisection of A077221.

[8*n^2-8*n+1: n in [0..50]]; // Vincenzo Librandi, Feb 05 2013

FoldList[#1 + #2 &, 1, 16 Range@ 45] (* Robert G. Wilson v, Feb 02 2011 *)
Rest[CoefficientList[Series[-x(1+14x+x^2)/(x-1)^3,{x,0,50}],x]]  (* Harvey P. Dale, Apr 22 2011 *)

a(n)=8*n^2-8*n+1 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = 8*n^2 - 8*n + 1.
a(n) = A035008(n-1) + 1. - Omar E. Pol, Apr 26 2008
a(n) = 16*n + a(n-1) - 16 with n > 1, a(1)=1. - Vincenzo Librandi, Aug 08 2010
G.f.: -x*(1+14*x+x^2) / (x-1)^3. - R. J. Mathar, Feb 04 2011
E.g.f.: (8*x^2 + 1)*exp(x). - G. C. Greubel, Jul 18 2017
a(n) = A056220(2n-1). - Bruce J. Nicholson, Aug 31 2017
Sum_{n>=1} 1/a(n) = Pi * tan(Pi/(2*sqrt(2))) / (4*sqrt(2)). - Vaclav Kotesovec, Jul 23 2019
From Amiram Eldar, Jun 21 2020: (Start)
Sum_{n>=1} a(n)/n! = 9*e - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 9/e - 1. (End)
Product_{n>=2} (a(n) - 1) / (a(n) + 1) = Pi/4. - Dimitris Valianatos, Jun 27 2020
a(n) = A016754(n-1) + 8*A000217(n-1). - Leo Tavares, Jul 19 2021

A101321 Table T(n,m) = 1 + nm(m+1)/2 read by antidiagonals: centered polygonal numbers.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 7, 7, 4, 1, 1, 11, 13, 10, 5, 1, 1, 16, 21, 19, 13, 6, 1, 1, 22, 31, 31, 25, 16, 7, 1, 1, 29, 43, 46, 41, 31, 19, 8, 1, 1, 37, 57, 64, 61, 51, 37, 22, 9, 1, 1, 46, 73, 85, 85, 76, 61, 43, 25, 10, 1, 1, 56, 91, 109, 113, 106, 91, 71, 49, 28, 11, 1, 1, 67

Offset: 0

Eugene McDonnell (eemcd(AT)mac.com), Dec 24 2004

Row n gives the centered figurate numbers of the n-gon.

Antidiagonal sums are in A101338.

			The upper left corner of the infinite array T is
|0| 1   1   1   1   1   1   1   1   1   1 ... A000012
|1| 1   2   4   7  11  16  22  29  37  46 ... A000124
|2| 1   3   7  13  21  31  43  57  73  91 ... A002061
|3| 1   4  10  19  31  46  64  85 109 136 ... A005448
|4| 1   5  13  25  41  61  85 113 145 181 ... A001844
|5| 1   6  16  31  51  76 106 141 181 226 ... A005891
|6| 1   7  19  37  61  91 127 169 217 271 ... A003215
|7| 1   8  22  43  71 106 148 197 253 316 ... A069099
|8| 1   9  25  49  81 121 169 225 289 361 ... A016754
|9| 1  10  28  55  91 136 190 253 325 406 ... A060544

Cf. A000217, A001263, A101338.

A101321 := proc(n,k)
        n*k*(k+1)/2+1 ;
end proc: # R. J. Mathar, Jul 28 2016

T[n_, m_] := 1 + n m (m + 1)/2;
Table[T[n - m, m], {n, 0, 12}, {m, n, 0, -1}] // Flatten (* Jean-François Alcover, Mar 23 2020 *)

T(n,m) = 1 + n*m*(m+1)/2 \\ Charles R Greathouse IV, Jul 28 2016

T(n,2) = A016777(n). T(n,3) = A016921(n). T(n,4) = A017281(n).
T(10,m) = A062786(m+1).
T(11,m) = A069125(m+1).
T(12,m) = A003154(m+1).
T(13,m) = A069126(m+1).
T(14,m) = A069127(m+1).
T(15,m) = A069128(m+1).
T(16,m) = A069129(m+1).
T(17,m) = A069130(m+1).
T(18,m) = A069131(m+1).
T(19,m) = A069132(m+1).
T(20,m) = A069133(m+1).
T(n+1,m) = T(n,m) + m*(m+1)/2. - Gary W. Adamson and Michel Marcus, Oct 13 2015

Edited by R. J. Mathar, Oct 21 2009

A069190 Centered 24-gonal numbers.

Original entry on oeis.org

1, 25, 73, 145, 241, 361, 505, 673, 865, 1081, 1321, 1585, 1873, 2185, 2521, 2881, 3265, 3673, 4105, 4561, 5041, 5545, 6073, 6625, 7201, 7801, 8425, 9073, 9745, 10441, 11161, 11905, 12673, 13465, 14281, 15121, 15985, 16873, 17785, 18721, 19681, 20665, 21673

Offset: 1

Terrel Trotter, Jr., Apr 10 2002

Sequence found by reading the line from 1, in the direction 1, 25, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. Semi-axis opposite to A135453 in the same spiral. - Omar E. Pol, Sep 16 2011

			a(5) = 241 because 12*5^2 - 12*5 + 1 = 300 - 60 + 1 = 241.

Ivan Panchenko, Table of n, a(n) for n = 1..1000
John Elias, Illustration: Odd Ordered Star Perimeters.
Eric Weisstein's World of Mathematics, Centered Polygonal Numbers.
R. Yin, J. Mu, and T. Komatsu, The p-Frobenius Number for the Triple of the Generalized Star Numbers, Preprints 2024, 2024072280. See p. 2.
Index entries for sequences related to centered polygonal numbers.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. centered k-gonal numbers with k=3..25: A005448, A001844, A005891, A003215, A069099, A016754, A060544, A062786, A069125, A003154, A069126, A069127, A069128, A069129, A069130, A069131, A069132, A069133, A069178, A069173, A069174, A069190, A262221.

FoldList[#1 + #2 &, 1, 24 Range@ 45] (* Robert G. Wilson v *)
Table[12n^2-12n+1,{n,50}] (* or *) LinearRecurrence[{3,-3,1},{1,25,73},50] (* Harvey P. Dale, Jul 17 2011 *)

a(n)=12*n^2-12*n+1 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = 12*n^2 - 12*n + 1.
a(n) = 24*n + a(n-1) - 24 with a(1)=1. - Vincenzo Librandi, Aug 08 2010
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(1)=1, a(2)=25, a(3)=73. - Harvey P. Dale, Jul 17 2011
G.f.: x*(1+22*x+x^2)/(1-x)^3. - Harvey P. Dale, Jul 17 2011
Binomial transform of [1, 24, 24, 0, 0, 0, ...] and Narayana transform (cf. A001263) of [1, 24, 0, 0, 0, ...]. - Gary W. Adamson, Jul 26 2011
From Amiram Eldar, Jun 21 2020: (Start)
Sum_{n>=1} 1/a(n) = Pi*tan(Pi/sqrt(6))/(4*sqrt(6)).
Sum_{n>=1} a(n)/n! = 13*e - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 13/e - 1. (End)
E.g.f.: exp(x)*(1 + 12*x^2) - 1. - Stefano Spezia, May 31 2022

More terms from Harvey P. Dale, Jul 17 2011

A063491 a(n) = (2n - 1)(3n^2 - 3n + 2)/2.

Original entry on oeis.org

1, 12, 50, 133, 279, 506, 832, 1275, 1853, 2584, 3486, 4577, 5875, 7398, 9164, 11191, 13497, 16100, 19018, 22269, 25871, 29842, 34200, 38963, 44149, 49776, 55862, 62425, 69483, 77054, 85156, 93807, 103025, 112828, 123234, 134261, 145927, 158250, 171248, 184939

Offset: 1

N. J. A. Sloane, Aug 01 2001

A triangle has sides of lengths 6*n-3, 6*n^2-6*n+4, and 6*n^2-6*n+7; for n>2 its area is 6*sqrt(a(n)^2 - 1). - J. M. Bergot, Aug 30 2013

[The source of this is using (n,n+1), (n+1,n+2), and (n+2,n+3) as (a,b) in the creation of three Pythagorean triangles with sides b^2-a^2, 2*a*b, and a^2+b^2. Combine the three respective sides to create a new larger triangle, then find its area. It is not simply working backwards from the sequence. As well, the sequence has this as its first comment to show that the numbers are actually doing something to find a solution.]

T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.

Harry J. Smith, Table of n, a(n) for n = 1..1000
T. P. Martin, Shells of atoms, Phys. Rep., 273 (1996), 199-241, eq. (10).
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

1/12*t*(2*n^3-3*n^2+n)+2*n-1 for t = 2, 4, 6, ... gives A049480, A005894, A063488, A001845, A063489, A005898, A063490, A057813, A063491, A005902, A063492, A005917, A063493, A063494, A063495, A063496.

Cf. A005448, A004188.

[(2*n-1)*(3*n^2 -3*n +2)/2: n in [1..30]]; // G. C. Greubel, Dec 01 2017

LinearRecurrence[{4,-6,4,-1},{1,12,50,133},40] (* Harvey P. Dale, Jun 05 2016 *)
Table[(2*n-1)*(3*n^2 -3*n +2)/2, {n,1,30}] (* G. C. Greubel, Dec 01 2017 *)

a(n) = { (2*n - 1)*(3*n^2 - 3*n + 2)/2 } \\ Harry J. Smith, Aug 23 2009

my(x='x+O('x^30)); Vec(serlaplace((-2 + 4*x + 9*x^2 + 6*x^3)*exp(x)/2 + 1)) \\ G. C. Greubel, Dec 01 2017

a <- c(0, 1, 9, 38, 110)
for(n in (length(a)+1):40)
  a[n] <- +4*a[n-1]-6*a[n-2]+4*a[n-3]-a[n-4]
a [Yosu Yurramendi, Sep 04 2013]

G.f.: x*(1+x)*(1+7*x+x^2)/(1-x)^4. - Colin Barker, Apr 20 2012
a(n) = +4*a(n-1) -6*a(n-2) +4*a(n-3) -1*a(n-4) n > 3, a(1)=1, a(2)=12, a(3)=50, a(4)=133. - Yosu Yurramendi, Sep 04 2013
E.g.f.: (-2 + 4*x + 9*x^2 + 6*x^3)*exp(x)/2 + 1. - G. C. Greubel, Dec 01 2017
From Bruce J. Nicholson, Jun 17 2020: (Start)
a(n) = A005448(n) * A005408(n-1).
a(n) = A004188(n) + A004188(n-1). (End)

A005918 Number of points on surface of square pyramid: 3*n^2 + 2 (n>0).

Original entry on oeis.org

1, 5, 14, 29, 50, 77, 110, 149, 194, 245, 302, 365, 434, 509, 590, 677, 770, 869, 974, 1085, 1202, 1325, 1454, 1589, 1730, 1877, 2030, 2189, 2354, 2525, 2702, 2885, 3074, 3269, 3470, 3677, 3890, 4109, 4334, 4565, 4802, 5045, 5294, 5549, 5810, 6077, 6350, 6629

Offset: 0

N. J. A. Sloane

Also coordination sequence of the 5-connected (or bnn) net = hexagonal net X integers.
Also (except for initial term) numbers of the form 3n^2+2 that are not squares. All numbers 3n^2+2 are == 2 (mod 3), and hence not squares. - Cino Hilliard, Mar 01 2003, modified by Franklin T. Adams-Watters, Jun 27 2014
If a 2-set Y and a 3-set Z are disjoint subsets of an n-set X then a(n-4) is the number of 4-subsets of X intersecting both Y and Z. - Milan Janjic, Sep 08 2007
Sums of three consecutive squares: (n - 2)^2 + (n - 1)^2 + n^2 for n > 1. - Keith Tyler, Aug 10 2010

			G.f. = 1 + 5*x + 14*x^2 + 29*x^3 + 50*x^4 + 77*x^5 + 110*x^6 + 149*x^7 + ...

H. S. M. Coxeter, Polyhedral numbers, in R. S. Cohen et al., editors, For Dirk Struik. Reidel, Dordrecht, 1974, pp. 25-35.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A. F. Wells, Three-Dimensional Nets and Polyhedra, Fig. 15.1 (e).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Branko Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #26.
Milan Janjic, Two Enumerative Functions.
Milan Janjic and Boris Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550, 2013. - From _N. J. A. Sloane_, Feb 13 2013
Milan Janjic and Boris Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014), Article 14.3.5.
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.
Reticular Chemistry Structure Resource (RCSR), The bnn tiling (or net).
B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985),4545-4558.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A120328, A206399.

Partial sums give A063488.

A005918:=-(z+1)*(z**2+z+1)/(z-1)**3; # Simon Plouffe in his 1992 dissertation.

Join[{1}, Table[Plus@@(Range[n, n + 2]^2), {n, 0, 49}]] (* Alonso del Arte, Oct 27 2012 *)
CoefficientList[Series[(1 - x^2) (1 - x^3)/(1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Aug 07 2014 *)
LinearRecurrence[{3,-3,1},{1,5,14,29},50] (* Harvey P. Dale, Dec 12 2015 *)

sq3nsqp2(n) = { for(x=1,n, y = 3*x*x+2; print1(y, ", ") ) }

{a(n) = 3*n^2 + 2 - (n==0)}; /* Michael Somos, Aug 07 2014 */

G.f.: (1 - x^2)*(1 - x^3)/(1 - x)^5 = (1+x)*(1+x+x^2)/(1-x)^3.
Euler transform of length 3 sequence [ 5, -1, -1]. - Michael Somos, Aug 07 2014
a(-n) = a(n) for all n in Z. - Michael Somos, Aug 07 2014
a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>3. - Colin Barker, Aug 07 2014
a(0) = 1; for n > 0, a(n) = A120328(n-1). - Doug Bell, Aug 18 2015
E.g.f.: (2+3*x+3*x^2)*exp(x)-1. - Robert Israel, Aug 18 2015
a(n) = A005448(n) + A005448(n+1), sum of 2 consecutive centered triangular numbers. - R. J. Mathar, Apr 28 2020
a(n) = (n - 1)^2 + n^2 + (n + 1)^2. - Charlie Marion, Aug 31 2021
From Amiram Eldar, Sep 14 2022: (Start)
Sum_{n>=0} 1/a(n) = coth(sqrt(2/3)*Pi)*Pi/(2*sqrt(6)) + 3/4.
Sum_{n>=0} (-1)^n/a(n) = cosech(sqrt(2/3)*Pi)*Pi/(2*sqrt(6)) + 3/4. (End)

A061777 Start with a single triangle; at n-th generation add a triangle at each vertex, allowing triangles to overlap; sequence gives total population of triangles at n-th generation.

Original entry on oeis.org

1, 4, 10, 22, 40, 70, 112, 178, 268, 406, 592, 874, 1252, 1822, 2584, 3730, 5260, 7558, 10624, 15226, 21364, 30574, 42856, 61282, 85852, 122710, 171856, 245578, 343876, 491326, 687928, 982834, 1376044, 1965862, 2752288, 3931930, 5504788

Offset: 0

N. J. A. Sloane, R. K. Guy, Jun 23 2001

From the definition, assign label value "1" to an origin triangle; at n-th generation add a triangle at each vertex. Each non-overlapping triangle will have the same label value as that of the predecessor triangle to which it is connected; for the overlapping ones, the label value will be the sum of the label values of predecessors. a(n) is the sum of all label values at the n-th generation. The triangle count is A005448. See illustration. For n >= 1, (a(n) - a(n-1))/3 is A027383. - Kival Ngaokrajang, Sep 05 2014

R. Reed, The Lemming Simulation Problem, Mathematics in School, 3 (#6, Nov. 1974), front cover and pp. 5-6.

Kival Ngaokrajang, Illustration of initial terms
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]
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,2).

Partial sums of A061776.

Cf. A005448, A027383.

seq(`if`(n::even, 21*2^(n/2) - 6*n-20, 30*2^((n-1)/2)-6*n-20),n=0..100); # Robert Israel, Sep 14 2014

Table[If[EvenQ[n],21 2^(n/2)-6n-20,30 2^((n-1)/2)-6(n-1)-26],{n,0,40}] (* Harvey P. Dale, Nov 06 2011 *)

a(n)=if(n%2, 30, 21)<<(n\2) - 6*n - 20 \\ Charles R Greathouse IV, Sep 19 2014

From Colin Barker, May 08 2012: (Start)
a(n) = 21*2^(n/2) - 6*n - 20 if n is even.
a(n) = 30*2^((n-1)/2) - 6*(n - 1) - 26 if n is odd.
a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + 2*a(n-4).
G.f.: (1 + 2*x)*(1 + x^2)/((1 - x)^2*(1 - 2*x^2)). (End)
From Robert Israel, Sep 14 2014: (Start)
a(n) = -20 - 6*n + (21 + 15*sqrt(2))*sqrt(2)^(n-2) + (21 - 15*sqrt(2))*(-sqrt(2))^(n-2).
a(n) = 2*a(n-2) + ((3*n-2)/(3*n-5))*(a(n-1)-2*a(n-3)). (End)
E.g.f.: 21*cosh(sqrt(2)*x) + 15*sqrt(2)*sinh(sqrt(2)*x) - 2*exp(x)*(10 + 3*x). - Stefano Spezia, Aug 13 2022

Corrected by T. D. Noe, Nov 08 2006

A161644 Number of ON states after n generations of cellular automaton based on triangles.

Original entry on oeis.org

0, 1, 4, 10, 16, 22, 34, 52, 64, 70, 82, 106, 136, 160, 190, 232, 256, 262, 274, 298, 328, 358, 400, 466, 532, 568, 598, 658, 742, 814, 892, 988, 1036, 1042, 1054, 1078, 1108, 1138, 1180, 1246, 1312, 1354, 1396, 1474, 1588, 1702, 1816, 1966, 2104, 2164, 2194

Offset: 0

David Applegate and N. J. A. Sloane, Jun 15 2009

nonn

Analog of A151723 and A151725, but here we are working on the hexagonal net where each triangular cell has three neighbors (meeting along its edges). A cell is turned ON if exactly one of its three neighbors is ON. An ON cell remains ON forever.
We start with a single ON cell.
There is a dual version where the triangular cells meet vertex-to-vertex. The counts are the same: the two versions are isomorphic. Reed (1974) uses the vertex-to-vertex version. See the two Sloane "Illustration" links below to compare the two versions.
It appears that a(n) is also the number of polytoothpicks added in a toothpick structure formed by V-toothpicks but starting with a Y-toothpick: a(n) = a(n-1)+(A182632(n)-A182632(n-1))/2. (Checked up to n=39.) - Omar E. Pol, Dec 07 2010 and R. J. Mathar, Dec 17 2010
It appears that the behavior is similar to A161206. - Omar E. Pol, Jan 15 2016
It would be nice to have a formula or recurrence.
If new triangles are required to always move outwards we get A295559 and A295560.
From Paul Cousin, May 23 2025: (Start)
This is ETA rule 242 (11110010 in binary):
  -----------------------------------------------
  |state of the cell            |1|1|1|1|0|0|0|0|
  |sum of the neighbors' states |3|2|1|0|3|2|1|0|
  |cell's next state            |1|1|1|1|0|0|1|0|
  ----------------------------------------------- (End)

R. Reed, The Lemming Simulation Problem, Mathematics in School, 3 (#6, Nov. 1974), front cover and pp. 5-6. [Describes the dual structure where new triangles are joined at vertices rather than edges.]
S. Ulam, On some mathematical problems connected with patterns of growth of figures, pp. 215-224 of R. E. Bellman, ed., Mathematical Problems in the Biological Sciences, Proc. Sympos. Applied Math., Vol. 14, Amer. Math. Soc., 1962. See Example 3.

Paul Cousin, Table of n, a(n) for n = 0..16385 (first 10001 terms from Rémy Sigrist)
David Applegate, The movie version
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
Paul Cousin, ETA rule 242
Paul Cousin, Triangular Automata Integer Sequences
Lucas Garron, first 64 steps
Lucas Garron, after 128 steps
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]
Rémy Sigrist, PARI program for A161644
N. J. A. Sloane, Illustration of first 7 generations of A161644 and A295560 (edge-to-edge version)
N. J. A. Sloane, Illustration of first 11 generations of A161644 and A295560 (vertex-to-vertex version) [Include the 6 cells marked x to get A161644(11), exclude them to get A295560(11).]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A151723, A151725, A147562, A161206, A161645, A139250, A160120, A161206, A182632, A182840, A250300, A295559, A295560.

PARI
```
\\ See Links section.
```

a(n) = (A182632(n) - 1)/2, n >= 1. - Omar E. Pol, Mar 07 2013

Edited by N. J. A. Sloane, Jan 10 2010 and Nov 27 2017

A069125 a(n) = (11n^2 - 11n + 2)/2.

Original entry on oeis.org

1, 12, 34, 67, 111, 166, 232, 309, 397, 496, 606, 727, 859, 1002, 1156, 1321, 1497, 1684, 1882, 2091, 2311, 2542, 2784, 3037, 3301, 3576, 3862, 4159, 4467, 4786, 5116, 5457, 5809, 6172, 6546, 6931, 7327, 7734, 8152, 8581, 9021, 9472, 9934, 10407, 10891, 11386, 11892

Offset: 1

Terrel Trotter, Jr., Apr 07 2002

Centered hendecagonal (11-gonal) numbers. - Omar E. Pol, Oct 03 2011

Numbers of the form (2*m+1)^2 + k*m*(m+1)/2: in this case is k=3. See also A254963. - Bruno Berselli, Feb 11 2015

			a(5)=111 because 111 = (11*5^2 - 11*5 + 2)/2 = (275 - 55 + 2)/2 = 222/2.

T. D. Noe, Table of n, a(n) for n = 1..1000
X. Acloque, Polynexus Numbers and other mathematical wonders. [broken link]
Leo Tavares, Illustration: Clipped Stars.
Eric Weisstein's World of Mathematics, Centered Polygonal Numbers.
Index entries for sequences related to centered polygonal numbers.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001263, A001844, A003215, A005448, A005891, A069099.

Cf. A000217, A003154, A152740, A254963.

FoldList[#1 + #2 &, 1, 11 Range@ 45] (* Robert G. Wilson v *)
Table[(11n^2-11n+2)/2,{n,60}] (* or *) LinearRecurrence[{3,-3,1},{1,12,34},60] (* Harvey P. Dale, Jun 25 2011 *)

a(n)=(11*n^2-11*n+2)/2 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = 1 + Sum_{j=0..n-1} (11*j). - Xavier Acloque, Oct 22 2003
Binomial transform of [1, 11, 11, 0, 0, 0, ...]; Narayana transform (A001263) of [1, 11, 0, 0, 0, ...]. - Gary W. Adamson, Dec 29 2007
a(n) = 11*n + a(n-1) - 11 with n > 1, a(1)=1. - Vincenzo Librandi, Aug 08 2010
G.f.: -x*(1+9*x+x^2)/(x-1)^3. - R. J. Mathar, Jun 05 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=1, a(1)=12, a(2)=34. - Harvey P. Dale, Jun 25 2011
a(n) = A152740(n-1) + 1. - Omar E. Pol, Oct 03 2011
From Amiram Eldar, Jun 21 2020: (Start)
Sum_{n>=1} 1/a(n) = 2*Pi*tan(sqrt(3/11)*Pi/2)/sqrt(33).
Sum_{n>=1} a(n)/n! = 13*e/2 - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 13/(2*e) - 1. (End)
a(n) = A003154(n) - A000217(n-1). - Leo Tavares, Mar 29 2022
E.g.f.: exp(x)*(1 + 11*x^2/2) - 1. - Elmo R. Oliveira, Oct 18 2024

More terms from Harvey P. Dale, Jun 25 2011

Name rewritten by Bruno Berselli, Feb 11 2015

cp's OEIS Frontend

A077043 "Three-quarter squares": a(n) = n^2 - A002620(n).

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A068601 a(n) = n^3 - 1.

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A069129 Centered 16-gonal numbers.

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A101321 Table T(n,m) = 1 + n*m*(m+1)/2 read by antidiagonals: centered polygonal numbers.

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A069190 Centered 24-gonal numbers.

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A063491 a(n) = (2*n - 1)*(3*n^2 - 3*n + 2)/2.

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A101321 Table T(n,m) = 1 + nm(m+1)/2 read by antidiagonals: centered polygonal numbers.

A063491 a(n) = (2n - 1)(3n^2 - 3n + 2)/2.

A069125 a(n) = (11n^2 - 11n + 2)/2.