A008794 -id:A008794 - cp's OEIS frontend

A001477 The nonnegative integers.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77

Offset: 0

N. J. A. Sloane

Although this is a list, and lists normally have offset 1, it seems better to make an exception in this case. - N. J. A. Sloane, Mar 13 2010
The subsequence 0,1,2,3,4 gives the known values of n such that 2^(2^n)+1 is a prime (see A019434, the Fermat primes). - N. J. A. Sloane, Jun 16 2010
Also: The identity map, defined on the set of nonnegative integers. The restriction to the positive integers yields the sequence A000027. - M. F. Hasler, Nov 20 2013
The number of partitions of 2n into exactly 2 parts. - Colin Barker, Mar 22 2015
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 8960 or 168.- Philippe A.J.G. Chevalier, Dec 29 2015
Partial sums give A000217. - Omar E. Pol, Jul 26 2018
First differences are A000012 (the "all 1's" sequence). - M. F. Hasler, May 30 2020
See A061579 for the transposed infinite square matrix, or triangle with rows reversed. - M. F. Hasler, Nov 09 2021
This is the unique sequence (a(n)) that satisfies the inequality a(n+1) > a(a(n)) for all n in N. This simple and surprising result comes from the 6th problem proposed by Bulgaria during the second day of the 19th IMO (1977) in Belgrade (see link and reference). - Bernard Schott, Jan 25 2023

			Triangular view:
   0
   1   2
   3   4   5
   6   7   8   9
  10  11  12  13  14
  15  16  17  18  19  20
  21  22  23  24  25  26  27
  28  29  30  31  32  33  34  35
  36  37  38  39  40  41  42  43  44
  45  46  47  48  49  50  51  52  53  54

Maurice Protat, Des Olympiades à l'Agrégation, suite vérifiant f(n+1) > f(f(n)), Problème 7, pp. 31-32, Ellipses, Paris 1997.

N. J. A. Sloane, Table of n, a(n) for n = 0..500000
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
David Corneth, Counting to 13999 visualized | showing changes per digit, YouTube video, 2019.
Hans Havermann, Table giving n and American English name for n, for 0 <= n <= 100999, without spaces or hyphens
Hans Havermann, American English number names to one million, without spaces or hyphens
The IMO Compendium, Problem 6, 19th IMO 1977.
Tanya Khovanova, Recursive Sequences
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
László Németh, The trinomial transform triangle, J. Int. Seqs., Vol. 21 (2018), Article 18.7.3. Also arXiv:1807.07109 [math.NT], 2018.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 12.
Eric Weisstein's World of Mathematics, Natural Number
Eric Weisstein's World of Mathematics, Nonnegative Integer
Index entries for "core" sequences
Index entries for sequences that are permutations of the natural numbers
Index entries for linear recurrences with constant coefficients, signature (2,-1).
Index to sequences related to Olympiads.

Cf. A000027 (n>=1).
Cf. A000012 (first differences).
Partial sums of A057427. - Jeremy Gardiner, Sep 08 2002
Cf. A038608 (alternating signs), A001787 (binomial transform).
Cf. A055112.
Cf. Boustrophedon transforms: A231179, A000737.
Cf. A245422.
Number of orbits of Aut(Z^7) as function of the infinity norm A000579, A154286, A102860, A002412, A045943, A115067, A008586, A008585, A005843, A000217.
When written as an array, the rows/columns are A000217, A000124, A152948, A152950, A145018, A167499, A166136, A167487... and A000096, A034856, A055998, A046691, A052905, A055999... (with appropriate offsets); cf. analogous lists for A000027 in A185787.
Cf. A000290.
Cf. A061579 (transposed matrix / reversed triangle).

a001477 = id
a001477_list = [0..]  -- Reinhard Zumkeller, May 07 2012

print([n for n in 0:280]) # Paul Muljadi, Apr 15 2024

Magma
```
[ n : n in [0..100]];
```
Maple
```
[ seq(n,n=0..100) ];
```

Table[n, {n, 0, 100}] (* Stefan Steinerberger, Apr 08 2006 *)
LinearRecurrence[{2, -1}, {0, 1}, 77] (* Robert G. Wilson v, May 23 2013 *)
CoefficientList[ Series[x/(x - 1)^2, {x, 0, 76}], x] (* Robert G. Wilson v, May 23 2013 *)
Range[0,100] (* Harvey P. Dale, Dec 29 2024 *)

PARI
```
A001477(n)=n /* first term is a(0) */
```

def a(n): return n
print([a(n) for n in range(78)]) # Michael S. Branicky, Nov 13 2022

a(n) = n.
a(0) = 0, a(n) = a(n-1) + 1.
G.f.: x/(1-x)^2.
Multiplicative with a(p^e) = p^e. - David W. Wilson, Aug 01 2001
When seen as array: T(k, n) = n + (k+n)*(k+n+1)/2. Main diagonal is 2*n*(n+1) (A046092), antidiagonal sums are n*(n+1)*(n+2)/2 (A027480). - Ralf Stephan, Oct 17 2004
Dirichlet generating function: zeta(s-1). - Franklin T. Adams-Watters, Sep 11 2005
E.g.f.: x*e^x. - Franklin T. Adams-Watters, Sep 11 2005
a(0)=0, a(1)=1, a(n) = 2*a(n-1) - a(n-2). - Jaume Oliver Lafont, May 07 2008
Alternating partial sums give A001057 = A000217 - 2*(A008794). - Eric Desbiaux, Oct 28 2008
a(n) = 2*A080425(n) + 3*A008611(n-3), n>1. - Eric Desbiaux, Nov 15 2009
a(n) = A007966(n)*A007967(n). - Reinhard Zumkeller, Jun 18 2011
a(n) = Sum_{k>=0} A030308(n,k)*2^k. - Philippe Deléham, Oct 20 2011
a(n) = 2*A028242(n-1) + (-1)^n*A000034(n-1). - R. J. Mathar, Jul 20 2012
a(n+1) = det(C(i+1,j), 1 <= i, j <= n), where C(n,k) are binomial coefficients. - Mircea Merca, Apr 06 2013
a(n-1) = floor(n/e^(1/n)) for n > 0. - Richard R. Forberg, Jun 22 2013
a(n) = A000027(n) for all n>0.
a(n) = floor(cot(1/(n+1))). - Clark Kimberling, Oct 08 2014
a(0)=0, a(n>0) = 2*z(-1)^[( |z|/z + 3 )/2] + ( |z|/z - 1 )/2 for z = A130472(n>0); a 1 to 1 correspondence between integers and naturals. - Adriano Caroli, Mar 29 2015
G.f. as triangle: x*(1 + (x^2 - 5*x + 2)*y + x*(2*x - 1)*y^2)/((1 - x)^3*(1 - x*y)^3). - Stefano Spezia, Jul 22 2025

A005843 The nonnegative even numbers: a(n) = 2n.

Original entry on oeis.org

0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120

Offset: 0

N. J. A. Sloane

-2, -4, -6, -8, -10, -12, -14, ... are the trivial zeros of the Riemann zeta function. - Vivek Suri (vsuri(AT)jhu.edu), Jan 24 2008
If a 2-set Y and an (n-2)-set Z are disjoint subsets of an n-set X then a(n-2) is the number of 2-subsets of X intersecting both Y and Z. - Milan Janjic, Sep 19 2007
A134452(a(n)) = 0; A134451(a(n)) = 2 for n > 0. - Reinhard Zumkeller, Oct 27 2007
Omitting the initial zero gives the number of prime divisors with multiplicity of product of terms of n-th row of A077553. - Ray Chandler, Aug 21 2003
A059841(a(n))=1, A000035(a(n))=0. - Reinhard Zumkeller, Sep 29 2008
(APSO) Alternating partial sums of (a-b+c-d+e-f+g...) = (a+b+c+d+e+f+g...) - 2*(b+d+f...), it appears that APSO(A005843) = A052928 = A002378 - 2*(A116471), with A116471=2*A008794. - Eric Desbiaux, Oct 28 2008
A056753(a(n)) = 1. - Reinhard Zumkeller, Aug 23 2009
Twice the nonnegative numbers. - Juri-Stepan Gerasimov, Dec 12 2009
The number of hydrogen atoms in straight-chain (C(n)H(2n+2)), branched (C(n)H(2n+2), n > 3), and cyclic, n-carbon alkanes (C(n)H(2n), n > 2). - Paul Muljadi, Feb 18 2010
For n >= 1; a(n) = the smallest numbers m with the number of steps n of iterations of {r - (smallest prime divisor of r)} needed to reach 0 starting at r = m. See A175126 and A175127. A175126(a(n)) = A175126(A175127(n)) = n. Example (a(4)=8): 8-2=6, 6-2=4, 4-2=2, 2-2=0; iterations has 4 steps and number 8 is the smallest number with such result. - Jaroslav Krizek, Feb 15 2010
For n >= 1, a(n) = numbers k such that arithmetic mean of the first k positive integers is not integer. A040001(a(n)) > 1. See A145051 and A040001. - Jaroslav Krizek, May 28 2010
Union of A179082 and A179083. - Reinhard Zumkeller, Jun 28 2010
a(k) is the (Moore lower bound on and the) order of the (k,4)-cage: the smallest k-regular graph having girth four: the complete bipartite graph with k vertices in each part. - Jason Kimberley, Oct 30 2011
For n > 0: A048272(a(n)) <= 0. - Reinhard Zumkeller, Jan 21 2012
Let n be the number of pancakes that have to be divided equally between n+1 children. a(n) is the minimal number of radial cuts needed to accomplish the task. - Ivan N. Ianakiev, Sep 18 2013
For n > 0, a(n) is the largest number k such that (k!-n)/(k-n) is an integer. - Derek Orr, Jul 02 2014
a(n) when n > 2 is also the number of permutations simultaneously avoiding 213, 231 and 321 in the classical sense which can be realized as labels on an increasing strict binary tree with 2n-1 nodes. See A245904 for more information on increasing strict binary trees. - Manda Riehl Aug 07 2014
It appears that for n > 2, a(n) = A020482(n) + A002373(n), where all sequences are infinite. This is consistent with Goldbach's conjecture, which states that every even number > 2 can be expressed as the sum of two prime numbers. - Bob Selcoe, Mar 08 2015
Number of partitions of 4n into exactly 2 parts. - Colin Barker, Mar 23 2015
Number of neighbors in von Neumann neighborhood. - Dmitry Zaitsev, Nov 30 2015
Unique solution b( ) of the complementary equation a(n) = a(n-1)^2 - a(n-2)*b(n-1), where a(0) = 1, a(1) = 3, and a( ) and b( ) are increasing complementary sequences. - Clark Kimberling, Nov 21 2017
Also the maximum number of non-attacking bishops on an (n+1) X (n+1) board (n>0). (Cf. A000027 for rooks and queens (n>3), A008794 for kings or A030978 for knights.) - Martin Renner, Jan 26 2020
Integer k is even positive iff phi(2k) > phi(k), where phi is Euler's totient (A000010) [see reference De Koninck & Mercier]. - Bernard Schott, Dec 10 2020
Number of 3-permutations of n elements avoiding the patterns 132, 213, 312 and also number of 3-permutations avoiding the patterns 213, 231, 321. See Bonichon and Sun. - Michel Marcus, Aug 20 2022
a(n) gives the y-value of the integral solution (x,y) of the Pellian equation x^2 - (n^2 + 1)*y^2 = 1. The x-value is given by 2*n^2 + 1 (see Tattersall). - Stefano Spezia, Jul 24 2025

			G.f. = 2*x + 4*x^2 + 6*x^3 + 8*x^4 + 10*x^5 + 12*x^6 + 14*x^7 + 16*x^8 + ...

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 2.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 28.
J.-M. De Koninck and A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 529a pp. 71 and 257, Ellipses, 2004, Paris.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 256.

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
Nicolas Bonichon and Pierre-Jean Morel, Baxter d-permutations and other pattern avoiding classes, arXiv:2202.12677 [math.CO], 2022.
David Callan, On Ascent, Repetition and Descent Sequences, arXiv:1911.02209 [math.CO], 2019.
Charles Cratty, Samuel Erickson, Frehiwet Negass, and Lara Pudwell, Pattern Avoidance in Double Lists, preprint, 2015.
Kevin Fagan, Drabble cartoon, Jun 15 1987: Intelligence Test
Adam M. Goyt and Lara K. Pudwell, Avoiding colored partitions of two elements in the pattern sense, arXiv preprint arXiv:1203.3786 [math.CO], 2012, J. Int. Seq. 15 (2012) # 12.6.2
Milan Janjic, Two Enumerative Functions
Tanya Khovanova, Recursive Sequences
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.
Nathan Sun, On d-permutations and Pattern Avoidance Classes, arXiv:2208.08506 [math.CO], 2022.
Eric Weisstein's World of Mathematics, Even Number
Eric Weisstein's World of Mathematics, Hamiltonian Cycle
Eric Weisstein's World of Mathematics, Riemann Zeta Function Zeros
Wikipedia, Alkane
Index entries for "core" sequences
Index entries for linear recurrences with constant coefficients, signature (2,-1).

a(n)=2*A001477(n). - Juri-Stepan Gerasimov, Dec 12 2009
Cf. A000027, A002061, A005408, A001358, A077553, A077554, A077555, A002024, A087112, A157888, A157889, A140811, A157872, A157909, A157910, A165900.
Moore lower bound on the order of a (k,g) cage: A198300 (square); rows: A000027 (k=2), A027383 (k=3), A062318 (k=4), A061547 (k=5), A198306 (k=6), A198307 (k=7), A198308 (k=8), A198309 (k=9), A198310 (k=10), A094626 (k=11); columns: A020725 (g=3), this sequence (g=4), A002522 (g=5), A051890 (g=6), A188377 (g=7). - Jason Kimberley, Oct 30 2011
Cf. A231200 (boustrophedon transform).

a005843 = (* 2)
a005843_list = [0, 2 ..]  -- Reinhard Zumkeller, Feb 11 2012

Magma
```
[ 2*n : n in [0..100]];
```

A005843 := n->2*n;
A005843:=2/(z-1)**2; # Simon Plouffe in his 1992 dissertation

Range[0,120,2] (* Harvey P. Dale, Aug 16 2011 *)

PARI
```
A005843(n) = 2*n
```

def a(n): return 2*n # Martin Gergov, Oct 20 2022

R
```
seq(0,200,2)
```

G.f.: 2*x/(1-x)^2.
E.g.f.: 2*x*exp(x). - Geoffrey Critzer, Aug 25 2012
G.f. with interpolated zeros: 2x^2/((1-x)^2 * (1+x)^2); e.g.f. with interpolated zeros: x*sinh(x). - Geoffrey Critzer, Aug 25 2012
Inverse binomial transform of A036289, n*2^n. - Joshua Zucker, Jan 13 2006
a(0) = 0, a(1) = 2, a(n) = 2a(n-1) - a(n-2). - Jaume Oliver Lafont, May 07 2008
a(n) = Sum_{k=1..n} floor(6n/4^k + 1/2). - Vladimir Shevelev, Jun 04 2009
a(n) = A034856(n+1) - A000124(n) = A000217(n) + A005408(n) - A000124(n) = A005408(n) - 1. - Jaroslav Krizek, Sep 05 2009
a(n) = Sum_{k>=0} A030308(n,k)*A000079(k+1). - Philippe Deléham, Oct 17 2011
Digit sequence 22 read in base n-1. - Jason Kimberley, Oct 30 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Dec 23 2011
a(n) = 2*n = Product_{k=1..2*n-1} 2*sin(Pi*k/(2*n)), n >= 0 (undefined product := 1). See an Oct 09 2013 formula contribution in A000027 with a reference. - Wolfdieter Lang, Oct 10 2013
From Ilya Gutkovskiy, Aug 19 2016: (Start)
Convolution of A007395 and A057427.
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2)/2 = (1/2)*A002162 = (1/10)*A016655. (End)
From Bernard Schott, Dec 10 2020: (Start)
Sum_{n>=1} 1/a(n)^2 = Pi^2/24 = A222171.
Sum_{n>=1} (-1)^(n+1)/a(n)^2 = Pi^2/48 = A245058. (End)

A001318 Generalized pentagonal numbers: m(3m - 1)/2, m = 0, +-1, +-2, +-3, ....

Original entry on oeis.org

0, 1, 2, 5, 7, 12, 15, 22, 26, 35, 40, 51, 57, 70, 77, 92, 100, 117, 126, 145, 155, 176, 187, 210, 222, 247, 260, 287, 301, 330, 345, 376, 392, 425, 442, 477, 495, 532, 551, 590, 610, 651, 672, 715, 737, 782, 805, 852, 876, 925, 950, 1001, 1027, 1080, 1107, 1162, 1190, 1247, 1276, 1335

Offset: 0

N. J. A. Sloane

Partial sums of A026741. - Jud McCranie; corrected by Omar E. Pol, Jul 05 2012
From R. K. Guy, Dec 28 2005: (Start)
"Conway's relation twixt the triangular and pentagonal numbers: Divide the triangular numbers by 3 (when you can exactly):
0 1 3 6 10 15 21 28 36 45 55 66 78 91 105 120 136 153 ...
0 - 1 2 .- .5 .7 .- 12 15 .- 22 26 .- .35 .40 .- ..51 ...
.....-.-.....+..+.....-..-.....+..+......-...-.......+....
"and you get the pentagonal numbers in pairs, one of positive rank and the other negative.
"Append signs according as the pair have the same (+) or opposite (-) parity.
"Then Euler's pentagonal number theorem is easy to remember:
"p(n-0) - p(n-1) - p(n-2) + p(n-5) + p(n-7) - p(n-12) - p(n-15) ++-- = 0^n
where p(n) is the partition function, the left side terminates before the argument becomes negative and 0^n = 1 if n = 0 and = 0 if n > 0.
"E.g. p(0) = 1, p(7) = p(7-1) + p(7-2) - p(7-5) - p(7-7) + 0^7 = 11 + 7 - 2 - 1 + 0 = 15."
(End)
The sequence may be used in order to compute sigma(n), as described in Euler's article. - Thomas Baruchel, Nov 19 2003
Number of levels in the partitions of n + 1 with parts in {1,2}.
a(n) is the number of 3 X 3 matrices (symmetrical about each diagonal) M = {{a, b, c}, {b, d, b}, {c, b, a}} such that a + b + c = b + d + b = n + 2, a,b,c,d natural numbers; example: a(3) = 5 because (a,b,c,d) = (2,2,1,1), (1,2,2,1), (1,1,3,3), (3,1,1,3), (2,1,2,3). - Philippe Deléham, Apr 11 2007
Also numbers a(n) such that 24*a(n) + 1 = (6*m - 1)^2 are odd squares: 1, 25, 49, 121, 169, 289, 361, ..., m = 0, +-1, +-2, ... . - Zak Seidov, Mar 08 2008
From Matthew Vandermast, Oct 28 2008: (Start)
Numbers n for which A000326(n) is a member of A000332. Cf. A145920.
This sequence contains all members of A000332 and all nonnegative members of A145919. For values of n such that n*(3*n - 1)/2 belongs to A000332, see A145919. (End)
Starting with offset 1 = row sums of triangle A168258. - Gary W. Adamson, Nov 21 2009
Starting with offset 1 = Triangle A101688 * [1, 2, 3, ...]. - Gary W. Adamson, Nov 27 2009
Starting with offset 1 can be considered the first in an infinite set generated from A026741. Refer to the array in A175005. - Gary W. Adamson, Apr 03 2010
Vertex number of a square spiral whose edges have length A026741. The two axes of the spiral forming an "X" are A000326 and A005449. The four semi-axes forming an "X" are A049452, A049453, A033570 and the numbers >= 2 of A033568. - Omar E. Pol, Sep 08 2011
A general formula for the generalized k-gonal numbers is given by n*((k - 2)*n - k + 4)/2, n=0, +-1, +-2, ..., k >= 5. - Omar E. Pol, Sep 15 2011
a(n) is the number of 3-tuples (w,x,y) having all terms in {0,...,n} and 2*w = 2*x + y. - Clark Kimberling, Jun 04 2012
Generalized k-gonal numbers are second k-gonal numbers and positive terms of k-gonal numbers interleaved, k >= 5. - Omar E. Pol, Aug 04 2012
a(n) is the sum of the largest parts of the partitions of n+1 into exactly 2 parts. - Wesley Ivan Hurt, Jan 26 2013
Conway's relation mentioned by R. K. Guy is a relation between triangular numbers and generalized pentagonal numbers, two sequences from different families, but as triangular numbers are also generalized hexagonal numbers in this case we have a relation between two sequences from the same family. - Omar E. Pol, Feb 01 2013
Start with the sequence of all 0's. Add n to each value of a(n) and the next n - 1 terms. The result is the generalized pentagonal numbers. - Wesley Ivan Hurt, Nov 03 2014
(6k + 1) | a(4k). (3k + 1) | a(4k+1). (3k + 2) | a(4k+2). (6k + 5) | a(4k+3). - Jon Perry, Nov 04 2014
Enge, Hart and Johansson proved: "Every generalised pentagonal number c >= 5 is the sum of a smaller one and twice a smaller one, that is, there are generalised pentagonal numbers a, b < c such that c = 2a + b." (see link theorem 5). - Peter Luschny, Aug 26 2016
The Enge, et al. result for c >= 5 also holds for c >= 2 if 0 is included as a generalized pentagonal number. That is, 2 = 2*1 + 0. - Michael Somos, Jun 02 2018
Suggestion for title, where n actually matches the list and b-file: "Generalized pentagonal numbers: k(n)*(3*k(n) - 1)/2, where k(n) = A001057(n) = [0, 1, -1, 2, -2, 3, -3, ...], n >= 0" - Daniel Forgues, Jun 09 2018 & Jun 12 2018
Generalized k-gonal numbers are the partial sums of the sequence formed by the multiples of (k - 4) and the odd numbers (A005408) interleaved, with k >= 5. - Omar E. Pol, Jul 25 2018
The last digits form a symmetric cycle of length 40 [0, 1, 2, 5, ..., 5, 2, 1, 0], i.e., a(n) == a(n + 40) (mod 10) and a(n) == a(40*k - n - 1) (mod 10), 40*k > n. - Alejandro J. Becerra Jr., Aug 14 2018
Only 2, 5, and 7 are prime. All terms are of the form k*(k+1)/6, where 3 | k or 3 | k+1. For k > 6, the value divisible by 3 must have another factor d > 2, which will remain after the division by 6. - Eric Snyder, Jun 03 2022
8*a(n) is the product of two even numbers one of which is n + n mod 2. - Peter Luschny, Jul 15 2022
a(n) is the dot product of [1, 2, 3, ..., n] and repeat[1, 1/2]. a(5) = 12 = [1, 2, 3, 4, 5] dot [1, 1/2, 1, 1/2, 1] = [1 + 1 + 3 + 2 + 5]. - Gary W. Adamson, Dec 10 2022
Every nonnegative number is the sum of four terms of this sequence [S. Realis]. - N. J. A. Sloane, May 07 2023
From Peter Bala, Jan 06 2025: (Start)
The sequence terms are the exponents in the expansions of the following infinite products:
1) Product_{n >= 1} (1 - s(n)*q^n) = 1 + q + q^2 + q^5 + q^7 + q^12 + q^15 + ..., where s(n) = (-1)^(1 + mod(n+1,3)).
2) Product_{n >= 1} (1 - q^(2*n))*(1 - q^(3*n))^2/((1 - q^n)*(1 - q^(6*n))) = 1 + q + q^2 + q^5 + q^7 + q^12 + q^15 + ....
3) Product_{n >= 1} (1 - q^n)*(1 - q^(4*n))*(1 - q^(6*n))^5/((1 - q^(2*n))*(1 - q^(3*n))*(1 - q^(12*n)))^2 = 1 - q + q^2 - q^5 - q^7 + q^12 - q^15 + q^22 + q^26 - q^35 + ....
4) Product_{n >= 1} (1 - q^(2*n))^13/((1 - (-1)^n*q^n)*(1 - q^(4*n)))^5 = 1 - 5*q + 7*q^2 - 11*q^5 + 13*q^7 - 17*q^12 + 19*q^15 - + .... See Oliver, Theorem 1.1. (End)

			G.f. = x + 2*x^2 + 5*x^3 + 7*x^4 + 12*x^5 + 15*x^6 + 22*x^7 + 26*x^8 + 35*x^9 + ...

Enoch Haga, A strange sequence and a brilliant discovery, chapter 5 of Exploring prime numbers on your PC and the Internet, first revised ed., 2007 (and earlier ed.), pp. 53-70.
Ross Honsberger, Ingenuity in Mathematics, Random House, 1970, p. 117.
Donald E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, (to appear), section 7.2.1.4, equation (18).
Ivan Niven and Herbert S. Zuckerman, An Introduction to the Theory of Numbers, 2nd ed., Wiley, NY, 1966, p. 231.
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..1000
G. E. Andrews and J. A. Sellers, Congruences for the Fishburn Numbers, arXiv preprint arXiv:1401.5345 [math.NT], 2014.
Paul Barry, On sequences with {-1, 0, 1} Hankel transforms, arXiv preprint arXiv:1205.2565 [math.CO], 2012. - From _N. J. A. Sloane_, Oct 18 2012
Burkard Polster (Mathologer), The hardest "What comes next?" (Euler's pentagonal formula), Youtube video, Oct 17 2020.
S. Cooper and M. D. Hirschhorn, Results of Hurwitz type for three squares, Discrete Math., Vol. 274, No. 1-3 (2004), pp. 9-24. See P(q).
Stephen Eberhart, Letter to N. J. A. Sloane, Jan 19 1978.
John Elias, Illustration of Initial Terms: Generalized Penthexagrams.
John Elias, Illustration: Star Number Fractal.
John Elias, Illustration: Generalized Pentagonals In Generalized-Octagonal-Hexagrams.
Andreas Enge, William Hart, and Fredrik Johansson, Short addition sequences for theta functions, arXiv:1608.06810 [math.NT], 2016.
Leonhard Euler, Découverte d'une loi tout extraordinaire des nombres par rapport à la somme de leurs diviseurs, Opera Omnia, Series I, Vol. 2 (1751), pp. 241-253.
Leonhard Euler, On the remarkable properties of the pentagonal numbers, arXiv:math/0505373 [math.HO], 2005.
Leonhard Euler, De mirabilibus proprietatibus numerorum pentagonalium, par. 2
Leonhard Euler, Observatio de summis divisorum p. 8.
Leonhard Euler, An observation on the sums of divisors, p. 8, arXiv:math/0411587 [math.HO], 2004.
Alex Fink, Richard K. Guy, and Mark Krusemeyer, Partitions with parts occurring at most thrice, Contrib. Discr. Math., Vol. 3, No. 2 (2008), pp. 76-114.
Silvia Heubach and Toufik Mansour, Counting rises, levels and drops in compositions, arXiv:math/0310197 [math.CO], 2003.
Alfred Hoehn, Illustration of initial terms.
Barbara H. Margolius, Permutations with inversions, J. Integ. Seq., Vol. 4 (2001), Article 01.2.4.
Johannes W. Meijer, Euler's Ship on the Pentagonal Sea, pdf and jpg.
Johannes W. Meijer and Manuel Nepveu, Euler's ship on the Pentagonal Sea, Acta Nova, Vol. 4, No. 1 (December 2008), pp. 176-187.
Mircea Merca, The Lambert series factorization theorem, The Ramanujan Journal, January 2017; DOI: 10.1007/s11139-016-9856-3.
Mircea Merca and Maxie D. Schmidt, New Factor Pairs for Factorizations of Lambert Series Generating Functions, arXiv:1706.02359 [math.CO], 2017. See Remark 2.2.
Mircea Merca, Euler's partition function in terms of 2-adic valuation, Bol. Soc. Mat. Mex. 30, 76 (2024). See p. 3.
Ivan Niven, Formal power series, Amer. Math. Monthly, Vol. 76, No. 8 (1969), pp. 871-889.
Robert J. Lemke Oliver, Eta quotients and theta functions, Advances in Mathematics, Vol. 241, Jul. 2013, pp. 1-17.
Vladimir Pletser, Congruence conditions on the number of terms in sums of consecutive squared integers equal to squared integers, arXiv:1409.7969 [math.NT], 2014.
Vladimir Pletser, Finding all squared integers expressible as the sum of consecutive squared integers using generalized Pell equation solutions with Chebyshev polynomials, arXiv preprint arXiv:1409.7972 [math.NT], 2014.
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.
S. Realis, Question 271, Nouv. Corresp. Math., 4 (1878) 27-29.
Steven J. Schlicker, Numbers Simultaneously Polygonal and Centered Polygonal, Mathematics Magazine, Vol. 84, No. 5 (December 2011), pp. 339-350.
André Weil, Two lectures on number theory, past and present, L'Enseign. Math., Vol. XX (1974), pp. 87-110; Oeuvres III, pp. 279-302.
Eric Weisstein's World of Mathematics, Pentagonal numbers, Partition Function P.
Eric Weisstein's World of Mathematics, Pentagonal Number Theorem.
Wikipedia, Pentagonal number theorem.
M. Wohlgemuth, Pentagon, Kartenhaus und Summenzerlegung.
Keke Zhang, Generalized Catalan numbers, arXiv:2011.09593 [math.CO], 2020.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A080995 (characteristic function), A026741 (first differences), A034828 (partial sums), A165211 (mod 2).
Cf. A000326 (pentagonal numbers), A005449 (second pentagonal numbers), A000217 (triangular numbers).
Indices of nonzero terms of A010815, i.e., the (zero-based) indices of 1-bits of the infinite binary word to which the terms of A068052 converge.
Union of A036498 and A036499.
Cf. A153384, A168258, A101688, A174739, A175005.
Cf. A074378, A057569, A057570, A007310.
Sequences of generalized k-gonal numbers: this sequence (k=5), A000217 (k=6), A085787 (k=7), A001082 (k=8), A118277 (k=9), A074377 (k=10), A195160 (k=11), A195162 (k=12), A195313 (k=13), A195818 (k=14), A277082 (k=15), A274978 (k=16), A303305 (k=17), A274979 (k=18), A303813 (k=19), A218864 (k=20), A303298 (k=21), A303299 (k=22), A303303 (k=23), A303814 (k=24), A303304 (k=25), A316724 (k=26), A316725 (k=27), A303812 (k=28), A303815 (k=29), A316729 (k=30).
Column 1 of A195152.
Squares in APs: A221671, A221672.
Cf. A054440, A260664, A260672.
Quadrisection: A049453(k), A033570(k), A033568(k+1), A049452(k+1), k >= 0.
Cf. A002620.

a:=[0,1,2,5];; for n in [5..60] do a[n]:=2*a[n-2]-a[n-4]+3; od; a; # Muniru A Asiru, Aug 16 2018

a001318 n = a001318_list !! n
a001318_list = scanl1 (+) a026741_list -- Reinhard Zumkeller, Nov 15 2015

[(6*n^2 + 6*n + 1 - (2*n + 1)*(-1)^n)/16 : n in [0..50]]; // Wesley Ivan Hurt, Nov 03 2014

[(3*n^2 + 2*n + (n mod 2) * (2*n + 1)) div 8: n in [0..70]]; // Vincenzo Librandi, Nov 04 2014

A001318 := -(1+z+z**2)/(z+1)**2/(z-1)**3; # Simon Plouffe in his 1992 dissertation; gives sequence without initial zero
A001318 := proc(n) (6*n^2+6*n+1)/16-(2*n+1)*(-1)^n/16 ; end proc: # R. J. Mathar, Mar 27 2011

Table[n*(n+1)/6, {n, Select[Range[0, 100], Mod[#, 3] != 1 &]}]
Select[Accumulate[Range[0,200]]/3,IntegerQ] (* Harvey P. Dale, Oct 12 2014 *)
CoefficientList[Series[x (1 + x + x^2) / ((1 + x)^2 (1 - x)^3), {x, 0, 70}], x] (* Vincenzo Librandi, Nov 04 2014 *)
LinearRecurrence[{1,2,-2,-1,1},{0,1,2,5,7},70] (* Harvey P. Dale, Jun 05 2017 *)
a[ n_] := With[{m = Quotient[n + 1, 2]}, m (3 m + (-1)^n) / 2]; (* Michael Somos, Jun 02 2018 *)

{a(n) = (3*n^2 + 2*n + (n%2) * (2*n + 1)) / 8}; /* Michael Somos, Mar 24 2011 */

{a(n) = if( n<0, n = -1-n); polcoeff( x * (1 - x^3) / ((1 - x) * (1-x^2))^2 + x * O(x^n), n)}; /* Michael Somos, Mar 24 2011 */

{a(n) = my(m = (n+1) \ 2); m * (3*m + (-1)^n) / 2}; /* Michael Somos, Jun 02 2018 */

def a(n):
    p = n % 2
    return (n + p)*(3*n + 2 - p) >> 3
print([a(n) for n in range(60)])  # Peter Luschny, Jul 15 2022

def A001318(n): return n*(n+1)-(m:=n>>1)*(m+1)>>1 # Chai Wah Wu, Nov 23 2024

@CachedFunction
def A001318(n):
    if n == 0 : return 0
    inc = n//2 if is_even(n) else n
    return inc + A001318(n-1)
[A001318(n) for n in (0..59)] # Peter Luschny, Oct 13 2012

Euler: Product_{n>=1} (1 - x^n) = Sum_{n=-oo..oo} (-1)^n*x^(n*(3*n - 1)/2).
A080995(a(n)) = 1: complement of A090864; A000009(a(n)) = A051044(n). - Reinhard Zumkeller, Apr 22 2006
Euler transform of length-3 sequence [2, 2, -1]. - Michael Somos, Mar 24 2011
a(-1 - n) = a(n) for all n in Z. a(2*n) = A005449(n). a(2*n - 1) = A000326(n). - Michael Somos, Mar 24 2011. [The extension of the recurrence to negative indices satisfies the signature (1,2,-2,-1,1), but not the definition of the sequence m*(3*m -1)/2, because there is no m such that a(-1) = 0. - Klaus Purath, Jul 07 2021]
a(n) = 3 + 2*a(n-2) - a(n-4). - Ant King, Aug 23 2011
Product_{k>0} (1 - x^k) = Sum_{k>=0} (-1)^k * x^a(k). - Michael Somos, Mar 24 2011
G.f.: x*(1 + x + x^2)/((1 + x)^2*(1 - x)^3).
a(n) = n*(n + 1)/6 when n runs through numbers == 0 or 2 mod 3. - Barry E. Williams
a(n) = A008805(n-1) + A008805(n-2) + A008805(n-3), n > 2. - Ralf Stephan, Apr 26 2003
Sequence consists of the pentagonal numbers (A000326), followed by A000326(n) + n and then the next pentagonal number. - Jon Perry, Sep 11 2003
a(n) = (6*n^2 + 6*n + 1)/16 - (2*n + 1)*(-1)^n/16; a(n) = A034828(n+1) - A034828(n). - Paul Barry, May 13 2005
a(n) = Sum_{k=1..floor((n+1)/2)} (n - k + 1). - Paul Barry, Sep 07 2005
a(n) = A000217(n) - A000217(floor(n/2)). - Pierre CAMI, Dec 09 2007
If n even a(n) = a(n-1) + n/2 and if n odd a(n) = a(n-1) + n, n >= 2. - Pierre CAMI, Dec 09 2007
a(n)-a(n-1) = A026741(n) and it follows that the difference between consecutive terms is equal to n if n is odd and to n/2 if n is even. Hence this is a self-generating sequence that can be simply constructed from knowledge of the first term alone. - Ant King, Sep 26 2011
a(n) = (1/2)*ceiling(n/2)*ceiling((3*n + 1)/2). - Mircea Merca, Jul 13 2012
a(n) = (A008794(n+1) + A000217(n))/2 = A002378(n) - A085787(n). - Omar E. Pol, Jan 12 2013
a(n) = floor((n + 1)/2)*((n + 1) - (1/2)*floor((n + 1)/2) - 1/2). - Wesley Ivan Hurt, Jan 26 2013
From Oskar Wieland, Apr 10 2013: (Start)
a(n) = a(n+1) - A026741(n),
a(n) = a(n+2) - A001651(n),
a(n) = a(n+3) - A184418(n),
a(n) = a(n+4) - A007310(n),
a(n) = a(n+6) - A001651(n)*3 = a(n+6) - A016051(n),
a(n) = a(n+8) - A007310(n)*2 = a(n+8) - A091999(n),
a(n) = a(n+10)- A001651(n)*5 = a(n+10)- A072703(n),
a(n) = a(n+12)- A007310(n)*3,
a(n) = a(n+14)- A001651(n)*7. (End)
a(n) = (A007310(n+1)^2 - 1)/24. - Richard R. Forberg, May 27 2013; corrected by Zak Seidov, Mar 14 2015; further corrected by Jianing Song, Oct 24 2018
a(n) = Sum_{i = ceiling((n+1)/2)..n} i. - Wesley Ivan Hurt, Jun 08 2013
G.f.: x*G(0), where G(k) = 1 + x*(3*k + 4)/(3*k + 2 - x*(3*k + 2)*(3*k^2 + 11*k + 10)/(x*(3*k^2 + 11*k + 10) + (k + 1)*(3*k + 4)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 16 2013
Sum_{n>=1} 1/a(n) = 6 - 2*Pi/sqrt(3). - Vaclav Kotesovec, Oct 05 2016
a(n) = Sum_{i=1..n} numerator(i/2) = Sum_{i=1..n} denominator(2/i). - Wesley Ivan Hurt, Feb 26 2017
a(n) = A000292(A001651(n))/A001651(n), for n>0. - Ivan N. Ianakiev, May 08 2018
a(n) = ((-5 + (-1)^n - 6n)*(-1 + (-1)^n - 6n))/96. - José de Jesús Camacho Medina, Jun 12 2018
a(n) = Sum_{k=1..n} k/gcd(k,2). - Pedro Caceres, Apr 23 2019
Quadrisection. For r = 0,1,2,3: a(r + 4*k) = 6*k^2 + sqrt(24*a(r) + 1)*k + a(r), for k >= 1, with inputs (k = 0) {0,1,2,5}. These are the sequences A049453(k), A033570(k), A033568(k+1), A049452(k+1), for k >= 0, respectively. - Wolfdieter Lang, Feb 12 2021
a(n) = a(n-4) + sqrt(24*a(n-2) + 1), n >= 4. - Klaus Purath, Jul 07 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = 6*(log(3)-1). - Amiram Eldar, Feb 28 2022
a(n) = A002620(n) + A008805(n-1). Gary W. Adamson, Dec 10 2022
E.g.f.: (x*(7 + 3*x)*cosh(x) + (1 + 5*x + 3*x^2)*sinh(x))/8. - Stefano Spezia, Aug 01 2024

A236104 Triangle read by rows: T(n,k), n >= 1, k >= 1, in which column k lists k copies of the positive squares in nondecreasing order, and the first element of column k is in row k(k+1)/2.

Original entry on oeis.org

1, 4, 9, 1, 16, 1, 25, 4, 36, 4, 1, 49, 9, 1, 64, 9, 1, 81, 16, 4, 100, 16, 4, 1, 121, 25, 4, 1, 144, 25, 9, 1, 169, 36, 9, 1, 196, 36, 9, 4, 225, 49, 16, 4, 1, 256, 49, 16, 4, 1, 289, 64, 16, 4, 1, 324, 64, 25, 9, 1, 361, 81, 25, 9, 1, 400, 81, 25, 9, 4

Offset: 1

Omar E. Pol, Jan 23 2014

These are the squares of the entries of the triangle in A235791: T(n,k) = (A235791(n,k))^2.
Row n has length A003056(n) hence the first element of column k is in row A000217(k).
Columns 1-3 (including the initial zeros) are A000290, A008794, A211547.
Also column k lists the partial sums of the k-th column of triangle A196020 which gives an identity for sigma.
Since all the elements of this sequence are squares, we can draw an illustration of the alternating sum of row n step by step, and a symmetric diagram for A000203, A024916, A004125; see example.
For more information about the diagram see A237593.

			Triangle begins:
    1;
    4;
    9,   1;
   16,   1;
   25,   4;
   36,   4,   1;
   49,   9,   1;
   64,   9,   1;
   81,  16,   4;
  100,  16,   4,   1;
  121,  25,   4,   1;
  144,  25,   9,   1;
  169,  36,   9,   1;
  196,  36,   9,   4;
  225,  49,  16,   4,   1;
  256,  49,  16,   4,   1;
  289,  64,  16,   4,   1;
  324,  64,  25,   9,   1;
  361,  81,  25,   9,   1;
  400,  81,  25,   9,   4;
  441, 100,  36,   9,   4,   1;
  484, 100,  36,  16,   4,   1;
  529, 121,  36,  16,   4,   1;
  576, 121,  49,  16,   4,   1;
  ...
For n = 6 the sum of all divisors of all positive integers <= 6 is [1] + [1+2] + [1+3] + [1+2+4] + [1+5] + [1+2+3+6] = 1 + 3 + 4 + 7 + 6 + 12 = 33. On the other hand the 6th row of triangle is 36, 4, 1, therefore the alternating row sum is 36 - 4 + 1 = 33, equaling the sum of all divisors of all positive integers <= 6.
Illustration of the alternating sum of the 6th row as the area of a polygon (or the number of cells), step by step, in the fourth quadrant:
.     _ _ _ _ _ _       _ _ _ _ _ _       _ _ _ _ _ _
.    |           |     |           |     |           |
.    |           |     |           |     |           |
.    |           |     |           |     |           |
.    |           |     |        _ _|     |          _|
.    |           |     |       |         |        _|
.    |_ _ _ _ _ _|     |_ _ _ _|         |_ _ _ _|
.
.          36           36 - 4 = 32     36 - 4 + 1 = 33
.
Then using this method we can draw a symmetric diagram for A000203, A024916, A004125, as shown below:
--------------------------------------------------
n     A000203  A024916            Diagram
--------------------------------------------------
.                         _ _ _ _ _ _ _ _ _ _ _ _
1        1        1      |_| | | | | | | | | | | |
2        3        4      |_ _|_| | | | | | | | | |
3        4        8      |_ _|  _|_| | | | | | | |
4        7       15      |_ _ _|    _|_| | | | | |
5        6       21      |_ _ _|  _|  _ _|_| | | |
6       12       33      |_ _ _ _|  _| |  _ _|_| |
7        8       41      |_ _ _ _| |_ _|_|    _ _|
8       15       56      |_ _ _ _ _|  _|     |* *
9       13       69      |_ _ _ _ _| |      _|* *
10      18       87      |_ _ _ _ _ _|  _ _|* * *
11      12       99      |_ _ _ _ _ _| |* * * * *
12      28      127      |_ _ _ _ _ _ _|* * * * *
.
The total number of cells in the first n set of symmetric regions of the diagram equals A024916(n). It appears that the total number of cells in the n-th set of symmetric regions of the diagram equals sigma(n) = A000203(n). Example: for n = 12 the 12th row of triangle is 144, 25, 9, 1, hence the alternating sums is 144 - 25 + 9 - 1 = 127. On the other hand we have that A000290(12) - A004125(12) = 144 - 17 = A024916(12) = 127, equaling the total number of cells in the diagram after 12 stages. The number of cells in the 12th set of symmetric regions of the diagram is sigma(12) = A000203(12) = 28. Note that in this case there is only one region. Finally, the number of *'s is A004125(12) = 17.
Note that the diagram is also the top view of the stepped pyramid described in A245092. - _Omar E. Pol_, Feb 12 2018

Michael De Vlieger, Table of n, a(n) for n = 1..10075 (rows 1 <= n <= 500).

Cf. A000203, A000217, A000290, A001227, A003056, A008794, A024916, A004125, A196020, A211343, A228813, A231345, A231347, A235791, A235794, A235799, A236106, A236112, A236540, A237270, A237591, A237593, A239660, A244050, A245092, A262626, A286000.

Table[Ceiling[(n + 1)/k - (k + 1)/2]^2, {n, 20}, {k, Floor[(Sqrt[8 n + 1] - 1)/2]}] // Flatten (* Michael De Vlieger, Feb 10 2018, after Hartmut F. W. Hoft at A235791 *)

from sympy import sqrt
import math
def T(n, k): return int(math.ceil((n + 1)/k - (k + 1)/2))
for n in range(1, 21): print([T(n, k)**2 for k in range(1, int(math.floor((sqrt(8*n + 1) - 1)/2)) + 1)]) # Indranil Ghosh, Apr 25 2017

Sum_{k=1..A003056(n)} (-1)^(k-1)*T(n,k) = A024916(n). [Although this was stated as a fact, as far as I can tell, no proof was known. However, Don Reble has recently found a proof, which will be added here soon. - N. J. A. Sloane, Nov 23 2020]

A000203(n) = Sum_{k=1..A003056(n)} (-1)^(k-1) * (T(n,k) - T(n-1,k)), assuming that T(k*(k+1)/2-1,k) = 0. - Omar E. Pol, Oct 10 2018

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)

A109613 Odd numbers repeated.

Original entry on oeis.org

1, 1, 3, 3, 5, 5, 7, 7, 9, 9, 11, 11, 13, 13, 15, 15, 17, 17, 19, 19, 21, 21, 23, 23, 25, 25, 27, 27, 29, 29, 31, 31, 33, 33, 35, 35, 37, 37, 39, 39, 41, 41, 43, 43, 45, 45, 47, 47, 49, 49, 51, 51, 53, 53, 55, 55, 57, 57, 59, 59, 61, 61, 63, 63, 65, 65, 67, 67, 69, 69, 71, 71, 73

Offset: 0

Reinhard Zumkeller, Aug 01 2005

The number of rounds in a round-robin tournament with n competitors. - A. Timothy Royappa, Aug 13 2011
Diagonal sums of number triangle A113126. - Paul Barry, Oct 14 2005
When partitioning a convex n-gon by all the diagonals, the maximum number of sides in resulting polygons is 2*floor(n/2)+1 = a(n-1) (from Moscow Olympiad problem 1950). - Tanya Khovanova, Apr 06 2008
The inverse values of the coefficients in the series expansion of f(x) = (1/2)*(1+x)*log((1+x)/(1-x)) lead to this sequence; cf. A098557. - Johannes W. Meijer, Nov 12 2009
From Reinhard Zumkeller, Dec 05 2009: (Start)
First differences: A010673; partial sums: A000982;
A059329(n) = Sum_{k = 0..n} a(k)*a(n-k);
A167875(n) = Sum_{k = 0..n} a(k)*A005408(n-k);
A171218(n) = Sum_{k = 0..n} a(k)*A005843(n-k);
A008794(n+2) = Sum_{k = 0..n} a(k)*A059841(n-k). (End)
Dimension of the space of weight 2n+4 cusp forms for Gamma_0(5). - Michael Somos, May 29 2013
For n > 4: a(n) = A230584(n) - A230584(n-2). - Reinhard Zumkeller, Feb 10 2015
The arithmetic function v+-(n,2) as defined in A290988. - Robert Price, Aug 22 2017
For n > 0, also the chromatic number of the (n+1)-triangular (Johnson) graph. - Eric W. Weisstein, Nov 17 2017
a(n-1), for n >= 1, is also the upper bound a_{up}(b), where b = 2*n + 1, in the first (top) row of the complete coach system Sigma(b) of Hilton and Pedersen [H-P]. All odd numbers <= a_{up}(b) of the smallest positive restricted residue system of b appear once in the first rows of the c(2*n+1) = A135303(n) coaches. If b is an odd prime a_{up}(b) is the maximum. See a comment in the proof of the quasi-order theorem of H-P, on page 263 ["Furthermore, every possible a_i < b/2 ..."]. For an example see below. - Wolfdieter Lang, Feb 19 2020
Satisfies the nested recurrence a(n) = a(a(n-2)) + 2*a(n-a(n-1)) with a(0) = a(1) = 1. Cf. A004001. - Peter Bala, Aug 30 2022
The binomial transform is 1, 2, 6, 16, 40, 96, 224, 512, 1152, 2560,.. (see A057711). - R. J. Mathar, Feb 25 2023

			G.f. = 1 + x + 3*x^2 + 3*x^3 + 5*x^4 + 5*x^5 + 7*x^6 + 7*x^7 + 9*x^8 + 9*x^9 + ...
Complete coach system for (a composite) b = 2*n + 1 = 33: Sigma(33) ={[1; 5], [5, 7, 13; 2, 1, 2]} (the first two rows are here 1 and 5, 7, 13), a_{up}(33) = a(15) = 15. But 15 is not in the reduced residue system modulo 33, so the maximal (odd) a number is 13. For the prime b = 31, a_{up}(31) = a(14) = 15 appears as maximum of the first rows. - _Wolfdieter Lang_, Feb 19 2020

Peter Hilton and Jean Pedersen, A Mathematical Tapestry: Demonstrating the Beautiful Unity of Mathematics, Cambridge University Press, 2010, 3rd printing 2012, pp. (260-281).

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Chromatic Number
Eric Weisstein's World of Mathematics, Johnson Graph
Eric Weisstein's World of Mathematics, Triangular Graph
Wikipedia, Round-robin tournament
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A000217, A063196, A110660, A186421, A186422, A211955, A230584, A290988.
Complement of A052928 with respect to the universe A004526. - Guenther Schrack, Aug 21 2018
First differences of A000982, A061925, A074148, A105343, A116940, and A179207. - Guenther Schrack, Aug 21 2018

a109613 = (+ 1) . (* 2) . (`div` 2)
a109613_list = 1 : 1 : map (+ 2) a109613_list
-- Reinhard Zumkeller, Oct 27 2012, Feb 21 2011

A109613:=n->2*floor(n/2)+1; seq(A109613(k), k=0..100); # Wesley Ivan Hurt, Oct 22 2013

Flatten@ Array[{2# - 1, 2# - 1} &, 37] (* Robert G. Wilson v, Jul 07 2012 *)
(# - Boole[EvenQ[#]] &) /@ Range[80] (* Alonso del Arte, Sep 11 2019 *)
With[{c=2*Range[0,40]+1},Riffle[c,c]] (* Harvey P. Dale, Jan 02 2020 *)

A109613(n)=n>>1<<1+1 \\ Charles R Greathouse IV, Feb 24 2011

def a(n) : return( len( CuspForms( Gamma0( 5), 2*n + 4, prec=1). basis())); # Michael Somos, May 29 2013

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

a(n) = 2*floor(n/2) + 1.
a(n) = A052928(n) + 1 = 2*A004526(n) + 1.
a(n) = A028242(n) + A110654(n).
a(n) = A052938(n-2) + A084964(n-2) for n > 1. - Reinhard Zumkeller, Aug 27 2005
G.f.: (1 + x + x^2 + x^3)/(1 - x^2)^2. - Paul Barry, Oct 14 2005
a(n) = 2*a(n-2) - a(n-4), a(0) = 1, a(1) = 1, a(2) = 3, a(3) = 3. - Philippe Deléham, Nov 03 2008
a(n) = A001477(n) + A059841(n). - Philippe Deléham, Mar 31 2009
a(n) = 2*n - a(n-1), with a(0) = 1. - Vincenzo Librandi, Nov 13 2010
a(n) = R(n, -2), where R(n, x) is the n-th row polynomial of A211955. a(n) = (-1)^n + 2*Sum_{k = 1..n} (-1)^(n - k - 2)*4^(k-1)*binomial(n+k, 2*k). Cf. A084159. - Peter Bala, May 01 2012
a(n) = A182579(n+1, n). - Reinhard Zumkeller, May 06 2012
G.f.: ( 1 + x^2 ) / ( (1 + x)*(x - 1)^2 ). - R. J. Mathar, Jul 12 2016
E.g.f.: x*exp(x) + cosh(x). - Ilya Gutkovskiy, Jul 12 2016
From Guenther Schrack, Sep 10 2018: (Start)
a(-n) = -a(n-1).
a(n) = A047270(n+1) - (2*n + 2).
a(n) = A005408(A004526(n)). (End)
a(n) = A000217(n) / A004526(n+1), n > 0. - Torlach Rush, Nov 10 2023

A052928 The even numbers repeated.

Original entry on oeis.org

0, 0, 2, 2, 4, 4, 6, 6, 8, 8, 10, 10, 12, 12, 14, 14, 16, 16, 18, 18, 20, 20, 22, 22, 24, 24, 26, 26, 28, 28, 30, 30, 32, 32, 34, 34, 36, 36, 38, 38, 40, 40, 42, 42, 44, 44, 46, 46, 48, 48, 50, 50, 52, 52, 54, 54, 56, 56, 58, 58, 60, 60, 62, 62, 64, 64, 66, 66, 68, 68, 70, 70, 72, 72

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

a(n) is also the binary rank of the complete graph K(n). - Alessandro Cosentino (cosenal(AT)gmail.com), Feb 07 2009
Let I=I_n be the n X n identity matrix and P=P_n be the incidence matrix of the cycle (1,2,3,...,n). Then, for n >= 6, a(n) is the number of (0,1) n X n matrices A <= P^(-1)+I+P having exactly two 1's in every row and column with perA=2. - Vladimir Shevelev, Apr 12 2010
a(n+2) is the number of symmetry allowed, linearly independent terms at n-th order in the series expansion of the (E+A)xe vibronic perturbation matrix, H(Q) (cf. Eisfeld & Viel). - Bradley Klee, Jul 21 2015
The arithmetic function v_2(n,1) as defined in A289187. - Robert Price, Aug 22 2017
For n > 1, also the chromatic number of the n X n white bishop graph. - Eric W. Weisstein, Nov 17 2017
For n > 2, also the maximum vertex degree of the n-polygon diagonal intersection graph. - Eric W. Weisstein, Mar 23 2018
For n >= 2, a(n+2) gives the minimum weight of a Boolean function of algebraic degree at most n-2 whose support contains n linearly independent elements. - Christof Beierle, Nov 25 2019

C. D. Godsil and G. Royle, Algebraic Graph Theory, Springer, 2001, page 181. - Alessandro Cosentino (cosenal(AT)gmail.com), Feb 07 2009
V. S. Shevelyov (Shevelev), Extension of the Moser class of four-line Latin rectangles, DAN Ukrainy, 3(1992),15-19.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
C. Beierle, A. Biryukov and A. Udovenko, On degree-d zero-sum sets of full rank, Cryptography and Communications, November 2019.
W. Eisfeld and A. Viel, Higher order (A+E)xe pseudo-Jahn-Teller coupling, J. Chem. Phys., 122, 204317 (2005).
Nathan Fox, Finding Linear-Recurrent Solutions to Hofstadter-Like Recurrences Using Symbolic Computation, arXiv:1609.06342 [math.NT], 2016.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 914
J. Sondow and E. W. Weisstein, MathWorld: Wallis Formula
Eric Weisstein's World of Mathematics, Chromatic Number
Eric Weisstein's World of Mathematics, Legendre-Gauss Quadrature
Eric Weisstein's World of Mathematics, Maximum Vertex Degree
Eric Weisstein's World of Mathematics, Polygon Diagonal Intersection Graph
Eric Weisstein's World of Mathematics, Random Matrix
Eric Weisstein's World of Mathematics, White Bishop Graph
Index entries for linear recurrences with constant coefficients, signature (1,1,-1)
Index entries for Molien series

Cf. A000034, A000124, A004001, A004526, A005843, A007590, A008619, A008794, A032766, A064455, A099392, A109613, A118266, A123684, A124356, A192442, A289187, A342819.
First differences: A010673; partial sums: A007590; partial sums of partial sums: A212964(n+1).
Complement of A109613 with respect to universe A004526. - Guenther Schrack, Dec 07 2017
Is first differences of A099392. Fixed point sequence: A005843. - Guenther Schrack, May 30 2019
For n >= 3, A329822(n) gives the minimum weight of a Boolean function of algebraic degree at most n-3 whose support contains n linearly independent elements. - Christof Beierle, Nov 25 2019

a052928 = (* 2) . flip div 2
a052928_list = 0 : 0 : map (+ 2) a052928_list
-- Reinhard Zumkeller, Jun 20 2015

[2*Floor(n/2) : n in [0..50]]; // Wesley Ivan Hurt, Sep 13 2014

spec := [S,{S=Union(Sequence(Prod(Z,Z)),Prod(Sequence(Z),Sequence(Z)))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);

Flatten[Table[{2n, 2n}, {n, 0, 39}]] (* Alonso del Arte, Jun 24 2012 *)
With[{ev=2Range[0,40]},Riffle[ev,ev]] (* Harvey P. Dale, May 08 2021 *)
Table[Round[n + 1/2], {n, -1, 72}] (* Ed Pegg Jr, Jul 28 2025 *)

a(n)=n\2*2 \\ Charles R Greathouse IV, Nov 20 2011

a(n) = 2*floor(n/2).
G.f.: 2*x^2/((-1+x)^2*(1+x)).
a(n) + a(n+1) + 2 - 2*n = 0.
a(n) = n - 1/2 + (-1)^n/2.
a(n) = n + Sum_{k=1..n} (-1)^k. - William A. Tedeschi, Mar 20 2008
a(n) = a(n-1) + a(n-2) - a(n-3). - R. J. Mathar, Feb 19 2010
a(n) = |A123684(n) - A064455(n)| = A032766(n) - A008619(n-1). - Jaroslav Krizek, Mar 22 2011
For n > 0, a(n) = floor(sqrt(n^2+(-1)^n)). - Francesco Daddi, Aug 02 2011
a(n) = Sum_{k>=0} A030308(n,k)*b(k) with b(0)=0 and b(k)=2^k for k>0. - Philippe Deléham, Oct 19 2011
a(n) = A109613(n) - 1. - M. F. Hasler, Oct 22 2012
a(n) = n - (n mod 2). - Wesley Ivan Hurt, Jun 29 2013
a(n) = a(a(n-1)) + a(n-a(n-1)) for n>2. - Nathan Fox, Jul 24 2016
a(n) = 2*A004526(n). - Filip Zaludek, Oct 28 2016
E.g.f.: x*exp(x) - sinh(x). - Ilya Gutkovskiy, Oct 28 2016
a(-n) = -a(n+1); a(n) = A005843(A004526(n)). - Guenther Schrack, Sep 11 2018
From Guenther Schrack, May 29 2019: (Start)
a(b(n)) = b(n) + ((-1)^b(n) - 1)/2 for any sequence b(n) of offset 0.
a(a(n)) = a(n), idempotent.
a(A086970(n)) = A124356(n-1) for n > 1.
a(A000124(n)) = A192447(n+1).
a(n)*a(n+1)/2 = A007590(n), also equals partial sums of a(n).
A007590(a(n)) = 2*A008794(n). (End)

More terms from James Sellers, Jun 05 2000

Removed duplicate of recurrence; corrected original recurrence and g.f. against offset - R. J. Mathar, Feb 19 2010

A005993 Expansion of (1+x^2)/((1-x)^2*(1-x^2)^2).

Original entry on oeis.org

1, 2, 6, 10, 19, 28, 44, 60, 85, 110, 146, 182, 231, 280, 344, 408, 489, 570, 670, 770, 891, 1012, 1156, 1300, 1469, 1638, 1834, 2030, 2255, 2480, 2736, 2992, 3281, 3570, 3894, 4218, 4579, 4940, 5340, 5740, 6181, 6622, 7106, 7590, 8119, 8648, 9224, 9800

Offset: 0

N. J. A. Sloane, Winston C. Yang (yang(AT)math.wisc.edu)

Alkane (or paraffin) numbers l(6,n).
Dimension of the space of homogeneous degree n polynomials in (x1, y1, x2, y2) invariant under permutation of variables x1<->y1, x2<->y2.
Also multidigraphs with loops on 2 nodes with n arcs (see A138107). - Vladeta Jovovic, Dec 27 1999
Euler transform of finite sequence [2,3,0,-1]. - Michael Somos, Mar 17 2004
a(n-2) is the number of plane partitions with trace 2. - Michael Somos, Mar 17 2004
With offset 4, a(n) is the number of bracelets with n beads, 3 of which are red, 1 of which is blue. For odd n, a(n) = C(n-1,3)/2. For even n, a(n) = C(n-1,3)/2 +(n-2)/4. For n >= 6, with K = (n-1)(n-2)/((n-5)(n-4)), for odd n, a(n) = K*a(n-2). For even n, a(n) = K*a(n-2) -(n-2)/(n-5). - Washington Bomfim, Aug 05 2008
Equals (1,2,3,4,...) convolved with (1,0,3,0,5,...). - Gary W. Adamson, Feb 16 2009
Equals row sums of triangle A177878.
Equals (1/2)*((1, 4, 10, 20, 35, 56, ...) + (1, 0, 2 0, 3, 0, 4, ...)).
From Ctibor O. Zizka, Nov 21 2014: (Start)
With offset 4, a(n) is the number of different patterns of the 2-color 4-partition of n.
P(n)_(k;t) gives the number of different patterns of the t-color, k-partition of n.
P(n)(k;t) = 1 + Sum(i=2..n) Sum(j=2..i) Sum(r=1..m) c(i,j)*v_r*F_r(X_1,...,X_i).
P(n;i;j) = Sum(r=1..m) c_(i,j)*v_r*F_r(X_1,...,X_i).
m partition number of i.
c_(i,j) number of different coloring patterns on the r-th form (X_1,...,X_i) of i-partition with j-colors.
v_r number of i-partitions of n of the r-th form (X_1,...,X_i).
F_r(X_1,...,X_i) number of different patterns of the r-th form i-partition of n.
Some simple results:
  P(1)(k;t)=1, P(2)(k;t)=2, P(3)(k;t)=4, P(4)(k;t)=11, etc.
  P(n;1;1) = P(n;n;n) = 1 for all n;
  P(n;2;2) = floor(n/2) (A004526);
  P(n;3;2) = (n*n - 2*n + n mod 2)/4 (A002620).
This sequence is a(n) = P(n;4;2).
2-coloring of 4-partition is (A,B,A,B) or (B,A,B,A).
Each 4-partition of n has one of the form (X_1,X_1,X_1,X_1),(X_1,X_1,X_1,X_2), (X_1,X_1,X_2,X_2),(X_1,X_1,X_2,X_3),(X_1,X_2,X_3,X_4).
The number of forms is m=5 which is the partition number of k=4.
Partition form (X_1,X_1,X_1,X_1) gives 1 pattern ((X_1A,X_1B,X_1A,X_1B), (X_1,X_1,X_1,X_2) gives 2 patterns, (X_1,X_1,X_2,X_2) gives 4 patterns, (X_1,X_1,X_2,X_3) gives 6 patterns and (X_1,X_2,X_3,X_4) gives 12 patterns.
Thus a(n) = P(n;4;2) = 1*1*v_1 + 1*2*v_2 + 1*4*v_3 + 1*6*v_4 + 1*12*v_5 where v_r is the number of different 4-partitions of the r-th form (X_1,X_2,X_3,X_4) for a given n.
Example:
The 4-partitions of 8 are (2,2,2,2), (1,1,1,5), (1,1,3,3), (1,1,2,4), and (1,2,2,3):
  (2,2,2,2) 1 pattern
  (1,1,1,5), (1,1,5,1) 2 patterns
  (1,1,3,3), (1,3,3,1), (3,1,1,3), (1,3,1,3) 4 patterns
  (1,1,2,4), (1,1,4,2), (1,2,1,4), (1,2,4,1), (1,4,1,2), (2,1,1,4) 6 patterns
  (2,2,1,3), (2,2,3,1), (2,1,2,3), (2,1,3,2), (2,3,2,1), (1,2,2,3) 6 patterns
Thus a(8) = P(8,4,2) = 1 + 2 + 4 + 6 + 6 = 19. (End)
a(n) = length of run n+2 of consecutive 1's in A254338. - Reinhard Zumkeller, Feb 27 2015
Take a chessboard of (n+2) X (n+2) unit squares in which the a1 square is black. a(n) is the number of composite squares having black unit squares on their vertices. - Ivan N. Ianakiev, Jul 19 2018
a(n) is the number of 1423-avoiding odd Grassmannian permutations of size n+2. Avoiding any of the patterns 2314 or 3412 gives the same sequence. - Juan B. Gil, Mar 09 2023

			a(2) = 6, since ( x1*y1, x2*y2, x1*x1+y1*y1, x2*x2+y2*y2, x1*x2+y1*y2, x1*y2+x2*y1 ) are a basis for homogeneous quadratic invariant polynomials.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
L. Smith, Polynomial Invariants of Finite Groups, A K Peters, 1995, p. 96.

T. D. Noe, Table of n, a(n) for n = 0..1000
M. Benoumhani and M. Kolli, Finite topologies and partitions, JIS 13 (2010) # 10.3.5, Lemma 6 3rd line.
Washington Bomfim, The 19 bracelets with 8 beads - one blue, three reds and four blacks. [From _Washington Bomfim_, Aug 05 2008]
T. M. Brown, On the unimodality of convolutions of sequences of binomial coefficients, arXiv:1810.08235 [math.CO] (2018).
Johann Cigler, Some remarks on Rogers-Szegö polynomials and Losanitsch's triangle, arXiv:1711.03340 [math.CO], 2017.
Dragomir Z. Djokovic, Poincaré series of some pure and mixed trace algebras of two generic matrices, arXiv:math/0609262 [math.AC], 2006. See Table 8.
Juan B. Gil and Jessica A. Tomasko, Pattern-avoiding even and odd Grassmannian permutations, arXiv:2207.12617 [math.CO], 2022.
Naihuan Jing, Kailash Misra, and Carla Savage, On multi-color partitions and the generalized Rogers-Ramanujan identities, arXiv:math/9907183 [math.CO], 1999.
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)
N. J. A. Sloane, Classic Sequences
L. Smith, Polynomial invariants of finite groups. A survey of recent developments. Bull. Amer. Math. Soc. (N.S.) 34 (1997), no. 3, 211-250. See page 218. MR1433171 (98i:13009).
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A177878.
Partial sums of A008794 (without 0). - Bruno Berselli, Aug 30 2013
Cf. A254338, A002260, A005408, A282011.
Cf. A008619, A005995, A018211, A018213, A062136.

Following Gary W. Adamson.
import Data.List (inits, intersperse)
a005993 n = a005994_list !! n
a005993_list = map (sum . zipWith (*) (intersperse 0 [1, 3 ..]) . reverse) $
                   tail $ inits [1..]
-- Reinhard Zumkeller, Feb 27 2015

I:=[1,2,6,10,19,28]; [n le 6 select I[n] else 2*Self(n-1)+Self(n-2)-4*Self(n-3)+Self(n-4)+2*Self(n-5)-Self(n-6): n in [1..60]]; // Vincenzo Librandi, Jul 19 2015

g := proc(n) local i; add(floor(i/2)^2,i=1..n+1) end: # Joseph S. Riel (joer(AT)k-online.com), Mar 22 2002
a:= n-> (Matrix([[1, 0$3, -1, -2]]).Matrix(6, (i,j)-> if (i=j-1) then 1 elif j=1 then [2, 1, -4, 1, 2, -1][i] else 0 fi)^n)[1,1]; seq (a(n), n=0..44); # Alois P. Heinz, Jul 31 2008

CoefficientList[Series[(1+x^2)/((1-x)^2*(1-x^2)^2),{x,0,44}],x]  (* Jean-François Alcover, Apr 08 2011 *)
LinearRecurrence[{2,1,-4,1,2,-1},{1,2,6,10,19,28},50] (* Harvey P. Dale, Feb 20 2012 *)

a(n)=polcoeff((1+x^2)/(1-x)^2/(1-x^2)^2+x*O(x^n),n)

a(n) = (binomial(n+3, n) + (1-n%2)*binomial((n+2)/2, n>>1))/2 \\ Washington Bomfim, Aug 05 2008

a = vector(50); a[1]=1; a[2]=2;
for(n=3, 50, a[n] = ((n+2)*a[n-2]+2*a[n-1]-n)/(n-2)); a \\ Gerry Martens, Jun 03 2018

def A005993():
    a, b, to_be = 0, 0, True
    while True:
        yield (a*(a*(2*a+9)+13)+b*(b+1)*(2*b+1)+6)//6
        if to_be: b += 1
        else: a += 1
        to_be = not to_be
a = A005993()
[next(a) for  in range(48)] # _Peter Luschny, May 04 2016

l(c, r) = 1/2 C(c+r-3, r) + 1/2 d(c, r), where d(c, r) is C((c + r - 3)/2, r/2) if c is odd and r is even, 0 if c is even and r is odd, C((c + r - 4)/2, r/2) if c is even and r is even, C((c + r - 4)/2, (r - 1)/2) if c is odd and r is odd.
G.f.: (1+x^2)/((1-x)^2*(1-x^2)^2) = (1+x^2)/((1+x)^2*(x-1)^4) = (1/(1-x)^4 +1/(1-x^2)^2)/2.
a(2n) = (n+1)(2n^2+4n+3)/3, a(2n+1) = (n+1)(n+2)(2n+3)/3. a(-4-n) = -a(n).
From Yosu Yurramendi, Sep 12 2008: (Start)
a(n+1) = a(n) + A008794(n+3) with a(1)=1.
a(n) = A027656(n) + 2*A006918(n).
a(n+2) = a(n) + A000982(n+2) with a(1)=1, a(2)=2. (End)
a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6). - Jaume Oliver Lafont, Dec 05 2008
a(n) = (n^3 + 6*n^2 + 11*n + 6)/12 + ((n+2)/4)[n even] (the bracket means that the second term is added if and only if n is even). - Benoit Jubin, Mar 31 2012
a(n) = (1/12)*n*(n+1)*(n+2) + (1/4)*(n+1)*(1/2)*(1-(-1)^n), with offset 1. - Yosu Yurramendi, Jun 20 2013
a(n) = Sum_{i=0..n+1} ceiling(i/2) * round(i/2) = Sum_{i=0..n+2} floor(i/2)^2. - Bruno Berselli, Aug 30 2013
a(n) = (n + 2)*(3*(-1)^n + 2*n^2 + 8*n + 9)/24. - Ilya Gutkovskiy, May 04 2016
Recurrence formula:  a(n) = ((n+2)*a(n-2)+2*a(n-1)-n)/(n-2), a(1)=1, a(2)=2. - Gerry Martens, Jun 10 2018
E.g.f.: exp(-x)*(6 - 3*x + exp(2*x)*(18 + 39*x + 18*x^2 + 2*x^3))/24. - Stefano Spezia, Feb 23 2020
a(n) = Sum_{j=0..n/2} binomial(c+2*j-1,2*j)*binomial(c+n-2*j-1,n-2*j) where c=2. For other values of c we have: A008619 (c=1), A005995 (c=3), A018211 (c=4), A018213 (c=5), A062136 (c=6). - Miquel A. Fiol, Sep 24 2024

A059260 Triangle read by rows giving coefficient T(i,j) of x^i y^j in 1/(1-y-x*y-x^2) = 1/((1+x)(1-x-y)) for (i,j) = (0,0), (1,0), (0,1), (2,0), (1,1), (0,2), ...

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 0, 2, 2, 1, 1, 2, 4, 3, 1, 0, 3, 6, 7, 4, 1, 1, 3, 9, 13, 11, 5, 1, 0, 4, 12, 22, 24, 16, 6, 1, 1, 4, 16, 34, 46, 40, 22, 7, 1, 0, 5, 20, 50, 80, 86, 62, 29, 8, 1, 1, 5, 25, 70, 130, 166, 148, 91, 37, 9, 1, 0, 6, 30, 95, 200, 296, 314, 239, 128, 46, 10, 1

Offset: 0

N. J. A. Sloane, Jan 23 2001

Coefficients of the (left, normalized) shifted cyclotomic polynomial. Or, coefficients of the basic n-th q-series for q=-2. Indeed, let Y_n(x) = Sum_{k=0..n} x^k, having as roots all the n-th roots of unity except for 0; then coefficients in x of (-1)^n Y_n(-x-1) give exactly the n-th row of A059260 and a practical way to compute it. - Olivier Gérard, Jul 30 2002
The maximum in the (2n)-th row is T(n,n), which is A026641; also T(n,n) ~ (2/3)*binomial(2n,n). The maximum in the (2n-1)-th row is T(n-1,n), which is A014300 (but T does not have the same definition as in A026637); also T(n-1,n) ~ (1/3)*binomial(2n,n). Here is a generalization of the formula given in A026641: T(i,j) = Sum_{k=0..j} binomial(i+k-x,j-k)*binomial(j-k+x,k) for all x real (the proof is easy by induction on i+j using T(i,j) = T(i-1,j) + T(i,j-1)). - Claude Morin, May 21 2002
The second greatest term in the (2n)-th row is T(n-1,n+1), which is A014301; the second greatest term in the (2n+1)-th row is T(n+1,n) = 2*T(n-1,n+1), which is 2*A014301. - Claude Morin
Diagonal sums give A008346. - Paul Barry, Sep 23 2004
Riordan array (1/(1-x^2), x/(1-x)). As a product of Riordan arrays, factors into the product of (1/(1+x),x) and (1/(1-x),1/(1-x)) (binomial matrix). - Paul Barry, Oct 25 2004
Signed version is A239473 with relations to partial sums of sequences. - Tom Copeland, Mar 24 2014
From Robert Coquereaux, Oct 01 2014: (Start)
Columns of the triangle (cf. Example below) give alternate partial sums along nw-se diagonals of the Pascal triangle, i.e., sequences A000035, A004526, A002620 (or A087811), A002623 (or A173196), A001752, A001753, A001769, A001779, A001780, A001781, A001786, A001808, etc.
The dimension of the space of closed currents (distributional forms) of degree p on Gr(n), the Grassmann algebra with n generators, equivalently, the dimension of the space of Gr(n)-valued symmetric multilinear forms with vanishing graded divergence, is V(n,p) = 2^n T(p,n-1) - (-1)^p.
If p is odd V(n,p) is also the dimension of the cyclic cohomology group of order p of the Z2 graded algebra Gr(n).
If p is even the dimension of this cohomology group is V(n,p)+1.
Cf. A193844. (End)
From Peter Bala, Feb 07 2024: (Start)
The following remarks assume the row indexing starts at n = 1.
The sequence of row polynomials R(n,x), beginning R(1,x) = 1, R(2,x) = x, R(3,x) = 1 + x + x^2 , ..., is a strong divisibility sequence of polynomials in the ring Z[x]; that is, for all positive integers n and m, poly_gcd( R(n,x), R(m,x)) = R(gcd(n, m), x) - apply Norfleet (2005), Theorem 3. Consequently, the polynomial sequence {R(n,x): n >= 1} is a divisibility sequence; that is, if n divides m then R(n,x) divides R(m,x) in Z[x]. (End)
From Miquel A. Fiol, Oct 04 2024: (Start)
For j>=1, T(i,j) is the independence number of the (i-j)-supertoken graph FF_(i-j)(S_j) of the star graph S_j with j points.
(Given a graph G on n vertices and an integer k>=1, the k-supertoken (or reduced k-th power) FF_k(G) of G has vertices representing configurations of k indistinguishable tokens in the (not necessarily different) vertices of G, with two configurations being adjacent if one can be obtained from the other by moving one token along an edge. See an example below.)
Following the suggestion of Peter Munn, the k-supertoken graph FF_k(S_j) can also be defined as follows: Consider the Lattice graph L(k,j), whose vertices are the k^j j-vectors with elements in the set {0,..,k-1}, two being adjacent if they differ in just one coordinate by one unity. Then, FF_k(S_j) is the subgraph of L(k+1,j) induced by the vertices at distance at most k from (0,..,0). (End)

			Triangle begins
  1;
  0,  1;
  1,  1,  1;
  0,  2,  2,  1;
  1,  2,  4,  3,  1;
  0,  3,  6,  7,  4,  1;
  1,  3,  9, 13, 11,  5,  1;
  0,  4, 12, 22, 24, 16,  6,  1;
  1,  4, 16, 34, 46, 40, 22,  7,  1;
  0,  5, 20, 50, 80, 86, 62, 29,  8,  1;
Sequences obtained with _Miquel A. Fiol_'s Sep 30 2024 formula of A(n,c1,c2) for other values of (c1,c2). (In the table, rows are indexed by c1=0..6 and columns by c2=0..6):
A000007  A000012  A000027  A025747  A000292* A000332* A000389*
A059841  A008619  A087811* A002623  A001752  A001753  A001769
A193356  A008794* A005993  A005994  -------  -------  -------
-------  -------  -------  A005995  A018210  -------  A052267
-------  -------  -------  -------  A018211  A018212  -------
-------  -------  -------  -------  -------  A018213  A018214
-------  -------  -------  -------  -------  -------  A062136
*requires offset adjustment.
The 2-supertoken FF_2(S_3) of the star graph S_3 with central vertex 1 and peripheral vertices 2,3,4. (The vertex `ij' of FF_2(S_3) represents the configuration of one token in `ì' and the other token in `j'). The T(5,3)=7 independent vertices are 22, 24, 44, 23, 11, 34, and 33.
     22--12---24---14---44
          | \    / |
         23   11   34
            \  |  /
              13
               |
              33

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Roland Bacher, Chebyshev polynomials, quadratic surds and a variation of Pascal's triangle, arXiv:1509.09054 [math.CO], 2015.
Joseph Briggs, Alex Parker, Coy Schwieder, and Chris Wells, Frogs, hats and common subsequences, arXiv:2404.07285 [math.CO], 2024. See p. 28.
Robert Coquereaux and Éric Ragoucy, Currents on Grassmann algebras, J. of Geometry and Physics, 1995, Vol 15, pp 333-352.
Robert Coquereaux and Éric Ragoucy, Currents on Grassmann algebras, arXiv:hep-th/9310147, 1993.
Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus. II. A compendium of results, arXiv:2305.01100 [math.CO], 2023. See p. 8.
R. H. Hammack and G. D. Smith, Cycle bases of reduced powers of graphs, Ars Math. Contemp. 12 (2017) 183-203.
Christian Kassel, A Künneth formula for the cyclic cohomology of Z2-graded algebras, Math. Ann. 275 (1986) 683.
Ana Filipa Loureiro and Pascal Maroni, Polynomial sequences associated with the classical linear functionals, Numerical Algorithms, June 2012, Volume 60, Issue 2, pp 297-314. - From _N. J. A. Sloane_, Oct 12 2012
Ana Filipa Loureiro and Pascal Maroni, Polynomial sequences associated with the classical linear functionals, preprint, Centro de Matemática da Universidade do Porto.
MathOverflow, Cyclotomic Polynomials in Combinatorics
Mark Norfleet, Characterization of second-order strong divisibility sequences of polynomials, The Fibonacci Quarterly, 43(2) (2005), 166-169.

Cf. A059259. Row sums give A001045.
Seen as a square array read by antidiagonals this is the coefficient of x^k in expansion of 1/((1-x^2)*(1-x)^n) with rows A002620, A002623, A001752, A001753, A001769, A001779, A001780, A001781, A001786, A001808 etc. (allowing for signs). A058393 would then effectively provide the table for nonpositive n. - Henry Bottomley, Jun 25 2001
Cf. A026641, A014300.
Cf. A007318, A097805, A097808, A130595, A200139, A062135.

read transforms; 1/(1-y-x*y-x^2); SERIES2(%,x,y,12); SERIES2TOLIST(%,x,y,12);

t[n_, k_] := Sum[ (-1)^(n-j)*Binomial[j, k], {j, 0, n}]; Flatten[ Table[t[n, k], {n, 0, 12}, {k, 0, n}]] (* Jean-François Alcover, Oct 20 2011, after Paul Barry *)

T(n, k) = sum(j=0, n, (-1)^(n - j)*binomial(j, k));
for(n=0, 12, for(k=0, n, print1(T(n, k),", ");); print();) \\ Indranil Ghosh, Apr 11 2017

from sympy import binomial
def T(n, k): return sum((-1)**(n - j)*binomial(j, k) for j in range(n + 1))
for n in range(13): print([T(n, k) for k in range(n + 1)]) # Indranil Ghosh, Apr 11 2017

def A059260_row(n):
    @cached_function
    def prec(n, k):
        if k==n: return 1
        if k==0: return 0
        return -prec(n-1,k-1)-sum(prec(n,k+i-1) for i in (2..n-k+1))
    return [(-1)^(n-k+1)*prec(n+1, n-k+1) for k in (1..n)]
for n in (1..9): print(A059260_row(n)) # Peter Luschny, Mar 16 2016

G.f.: 1/(1-y-x*y-x^2) = 1 + y + x^2 + xy + y^2 + 2x^2y + 2xy^2 + y^3 + ...
E.g.f: (exp(-t)+(x+1)*exp((x+1)*t))/(x+2). - Tom Copeland, Mar 19 2014
O.g.f. (n-th row): ((-1)^n+(x+1)^(n+1))/(x+2). - Tom Copeland, Mar 19 2014
T(i, 0) = 1 if i is even or 0 if i is odd, T(0, i) = 1 and otherwise T(i, j) = T(i-1, j) + T(i, j-1); also T(i, j) = Sum_{m=j..i+j} (-1)^(i+j+m)*binomial(m, j). - Robert FERREOL, May 17 2002
T(i, j) ~ (i+j)/(2*i+j)*binomial(i+j, j); more precisely, abs(T(i, j)/binomial(i+j, j) - (i+j)/(2*i+j) )<=1/(4*(i+j)-2); the proof is by induction on i+j using the formula 2*T(i, j) = binomial(i+j, j)+T(i, j-1). - Claude Morin, May 21 2002
T(n, k) = Sum_{j=0..n} (-1)^(n-j)binomial(j, k). - Paul Barry, Aug 25 2004
T(n, k) = Sum_{j=0..n-k} binomial(n-j, j)*binomial(j, n-k-j). - Paul Barry, Jul 25 2005
Equals A097807 * A007318. - Gary W. Adamson, Feb 21 2007
Equals A128173 * A007318 as infinite lower triangular matrices. - Gary W. Adamson, Feb 17 2007
Equals A130595*A097805*A007318 = (inverse Pascal matrix)*(padded Pascal matrix)*(Pascal matrix) = A130595*A200139. Inverse is A097808 = A130595*(padded A130595)*A007318. - Tom Copeland, Nov 14 2016
T(i, j) = binomial(i+j, j)-T(i-1, j). - Laszlo Major, Apr 11 2017
Recurrence for row polynomials (with row indexing starting at n = 1): R(n,x) = x*R(n-1,x) + (x + 1)*R(n-2,x) with R(1,x) = 1 and R(2,x) = x. - Peter Bala, Feb 07 2024
From Miquel A. Fiol, Sep 30 2024: (Start)
The triangle can be seen as a slice of a 3-dimensional table that links it to well-known sequences as follows.
The j-th column of the triangle, T(i,j) for i >= j, equals A(n,c1,c2) = Sum_{k=0..floor(n/2)} binomial(c1+2*k-1,2*k)*binomial(c2+n-2*k-1,n-2*k) when c1=1, c2=j, and n=i-j.
This gives T(i,j) = Sum_{k=0..floor((i-j)/2)} binomial(i-2*k-1, j-1). For other values of (c1,c2), see the example below. (End)

Formula corrected by Philippe Deléham, Jan 11 2014

cp's OEIS Frontend

A172049 Irregular triangle T(n,k) = 2k-1 with A008794(n+2) values in row n.

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A001477 The nonnegative integers.

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A005843 The nonnegative even numbers: a(n) = 2n.

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A001318 Generalized pentagonal numbers: m*(3*m - 1)/2, m = 0, +-1, +-2, +-3, ....

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A236104 Triangle read by rows: T(n,k), n >= 1, k >= 1, in which column k lists k copies of the positive squares in nondecreasing order, and the first element of column k is in row k(k+1)/2.

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

A001318 Generalized pentagonal numbers: m(3m - 1)/2, m = 0, +-1, +-2, +-3, ....