A000212 -id:A000212 - cp's OEIS frontend

A001840 Expansion of g.f. x/((1 - x)^2*(1 - x^3)).

0, 1, 2, 3, 5, 7, 9, 12, 15, 18, 22, 26, 30, 35, 40, 45, 51, 57, 63, 70, 77, 84, 92, 100, 108, 117, 126, 135, 145, 155, 165, 176, 187, 198, 210, 222, 234, 247, 260, 273, 287, 301, 315, 330, 345, 360, 376, 392, 408, 425, 442, 459, 477, 495, 513, 532, 551, 570, 590

Offset: 0

N. J. A. Sloane

a(n-3) is the number of aperiodic necklaces (Lyndon words) with 3 black beads and n-3 white beads.
Number of triangular partitions (see Almkvist).
Consists of arithmetic progression quadruples of common difference n+1 starting at A045943(n). Refers to the least number of coins needed to be rearranged in order to invert the pattern of a (n+1)-rowed triangular array. For instance, a 5-rowed triangular array requires a minimum of a(4)=5 rearrangements (shown bracketed here) for it to be turned upside down.
  .....{*}..................{*}*.*{*}{*}
  .....*.*....................*.*.*.{*}
  ....*.*.*....---------\......*.*.*
  ..{*}*.*.*...---------/.......*.*
  {*}{*}*.*{*}..................{*}
  - Lekraj Beedassy, Oct 13 2003
Partial sums of 1,1,1,2,2,2,3,3,3,4,4,4,... - Jon Perry, Mar 01 2004
Sum of three successive terms is a triangular number in natural order starting with 3: a(n)+a(n+1)+a(n+2) = T(n+2) = (n+2)*(n+3)/2. - Amarnath Murthy, Apr 25 2004
Apply Riordan array (1/(1-x^3),x) to n. - Paul Barry, Apr 16 2005
Absolute values of numbers that appear in A145919. - Matthew Vandermast, Oct 28 2008
In the Moree definition, (-1)^n*a(n) is the 3rd Witt transform of A033999 and (-1)^n*A004524(n) with 2 leading zeros dropped is the 2nd Witt transform of A033999. - R. J. Mathar, Nov 08 2008
Column sums of:
1 2 3 4 5 6 7  8  9.....
      1 2 3 4  5  6.....
            1  2  3.....
........................
----------------------
1 2 3 5 7 9 12 15 18  - Jon Perry, Nov 16 2010
a(n) is the sum of the positive integers <= n that have the same residue modulo 3 as n. They are the additive counterpart of the triple factorial numbers. - Peter Luschny, Jul 06 2011
a(n+1) is the number of 3-tuples (w,x,y) with all terms in {0,...,n} and w=3*x+y. - Clark Kimberling, Jun 04 2012
a(n+1) is the number of pairs (x,y) with x and y in {0,...,n}, x-y = (1 mod 3), and x+y < n. - Clark Kimberling, Jul 02 2012
a(n+1) is the number of partitions of n into two sorts of part(s) 1 and one sort of (part) 3. - Joerg Arndt, Jun 10 2013
Arrange A004523 in rows successively shifted to the right two spaces and sum the columns:
1  2  2  3  4  4  5  6  6...
      1  2  2  3  4  4  5...
            1  2  2  3  4...
                  1  2  2...
                        1...
------------------------------
1  2  3  5  7  9 12 15 18... - L. Edson Jeffery, Jul 30 2014
a(n) = A258708(n+1,1) for n > 0. - Reinhard Zumkeller, Jun 23 2015
Also the number of triples of positive integers summing to n + 4, the first less than each of the other two. Also the number of triples of positive integers summing to n + 2, the first less than or equal to each of the other two. - Gus Wiseman, Oct 11 2020
Also the lower matching number of the (n+1)-triangular honeycomb king graph = n-triangular grid graph (West convention). - Eric W. Weisstein, Dec 14 2024

			G.f. = x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 9*x^6 + 12*x^7 + 15*x^8 + 18*x^9 + ...
1+2+3=6=t(3), 2+3+5=t(4), 5+7+9=t(5).
[n] a(n)
--------
[1] 1
[2] 2
[3] 3
[4] 1 + 4
[5] 2 + 5
[6] 3 + 6
[7] 1 + 4 + 7
[8] 2 + 5 + 8
[9] 3 + 6 + 9
a(7) = floor(2/3) +floor(3/3) +floor(4/3) +floor(5/3) +floor(6/3) +floor(7/3) +floor(8/3) +floor(9/3) = 12. - _Bruno Berselli_, Aug 29 2013

Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 73, problem 25.
Ulrich Faigle, Review of Gerhard Post and G.J. Woeginger, Sports tournaments, home-away assignments and the break minimization problem, MR2224983(2007b:90134), 2007.
Hansraj Gupta, Partitions of j-partite numbers into twelve or a smaller number of parts. Collection of articles dedicated to Professor P. L. Bhatnagar on his sixtieth birthday. Math. Student 40 (1972), 401-441 (1974).
Richard K. Guy, A problem of Zarankiewicz, in P. Erdős and G. Katona, editors, Theory of Graphs (Proceedings of the Colloquium, Tihany, Hungary), Academic Press, NY, 1968, pp. 119-150, (p. 126, divided by 2).
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
Chwas Ahmed, Paul Martin, and Volodymyr Mazorchuk, On the number of principal ideals in d-tonal partition monoids, arXiv preprint arXiv:1503.06718 [math.CO], 2015.
Gert Almkvist, Asymptotic formulas and generalized Dedekind sums, Exper. Math., 7 (No. 4, 1998), pp. 343-359.
David J. Broadhurst, On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory, arXiv:hep-th/9604128, 1996.
David Broadhurst and Xavier Roulleau, Number of partitions of modular integers, arXiv:2502.19523 [math.NT], 2025. See p. 19.
Cristian Cobeli, Aaditya Raghavan, and Alexandru Zaharescu, On the central ball in a translation invariant involutive field, arXiv:2408.01864 [math.NT], 2024. See p. 7.
Neville de Mestre and John Baker, Pebbles, Ducks and Other Surprises, Australian Maths. Teacher, Vol. 48, No 3, 1992, pp. 4-7.
Peter M. Chema, Illustration of first 27 terms as corners of a double hexagon spiral from 0
H. Gupta, Partitions of j-partite numbers into twelve or a smaller number of parts, Math. Student 40 (1972), 401-441 (1974). [Annotated scanned copy]
Richard K. Guy, A problem of Zarankiewicz, Research Paper No. 12, Dept. of Math., Univ. Calgary, Jan. 1967. [Annotated and scanned copy, with permission]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 207.
Clark Kimberling, A Combinatorial Classification of Triangle Centers on the Line at Infinity, J. Int. Seq., Vol. 22 (2019), Article 19.5.4.
Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.
R. P. Loh, A. G. Shannon, and A. F. Horadam, Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients, Preprint, 1980.
Pieter Moree, The formal series Witt transform, Discr. Math. no. 295 vol. 1-3 (2005) 143-160.
Brian O'Sullivan and Thomas Busch, Spontaneous emission in ultra-cold spin-polarised anisotropic Fermi seas, arXiv 0810.0231v1 [quant-ph], 2008. [Eq 8a, lambda=3]
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.
Gerhard Post and G. J. Woeginger, Sports tournaments, home-away assignments and the break minimization problem, Discrete Optimization 3, pp. 165-173, 2006.
Michael Somos, Somos Polynomials.
Gary E. Stevens, A Connell-Like Sequence, J. Integer Seqs., 1 (1998), Article 98.1.4.
Andrei K. Svinin, On some class of sums, arXiv:1610.05387 [math.CO], 2016. See p. 7.
Eric Weisstein's World of Mathematics, Lower Matching Number.
Eric Weisstein's World of Mathematics, Triangular Grid Graph.
Eric Weisstein's World of Mathematics, Triangular Honeycomb King Graph.
Index entries for sequences related to Lyndon words.
Index entries for two-way infinite sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Cf. A000031, A001037, A008748, A051168.
Ordered union of triangular matchstick numbers A045943 and generalized pentagonal numbers A001318.
Cf. A058937.
A column of triangle A011847.
Cf. A000217, A130205.
Cf. A258708.
A001399 counts 3-part partitions, ranked by A014612.
A337483 counts either weakly increasing or weakly decreasing triples.
A337484 counts neither strictly increasing nor strictly decreasing triples.
A014311 ranks 3-part compositions, with strict case A337453.
Cf. A007052, A007304, A011782, A069905, A156040, A218004.

a001840 n = a001840_list !! n
a001840_list = scanl (+) 0 a008620_list
-- Reinhard Zumkeller, Apr 16 2012

[ n le 2 select n else n*(n+1)/2-Self(n-1)-Self(n-2): n in [1..58] ];  // Klaus Brockhaus, Oct 01 2009

A001840 := n->floor((n+1)*(n+2)/6);
A001840:=-1/((z**2+z+1)*(z-1)**3); # conjectured (correctly) by Simon Plouffe in his 1992 dissertation
seq(floor(binomial(n-1,2)/3), n=3..61); # Zerinvary Lajos, Jan 12 2009
A001840 :=  n -> add(k, k = select(k -> k mod 3 = n mod 3, [$1 .. n])): seq(A001840(n), n = 0 .. 58); # Peter Luschny, Jul 06 2011

a[0]=0; a[1]=1; a[n_]:= a[n]= n(n+1)/2 -a[n-1] -a[n-2]; Table[a[n], {n,0,100}]
f[n_] := Floor[(n + 1)(n + 2)/6]; Array[f, 59, 0] (* Or *)
CoefficientList[ Series[ x/((1 + x + x^2)*(1 - x)^3), {x, 0, 58}], x] (* Robert G. Wilson v *)
a[ n_] := With[{m = If[ n < 0, -3 - n, n]}, SeriesCoefficient[ x /((1 - x^3) (1 - x)^2), {x, 0, m}]]; (* Michael Somos, Jul 11 2011 *)
LinearRecurrence[{2,-1,1,-2,1},{0,1,2,3,5},60] (* Harvey P. Dale, Jul 25 2011 *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n+4,{3}],#[[1]]<#[[2]]&&#[[1]]<#[[3]]&]],{n,0,15}] (* Gus Wiseman, Oct 05 2020 *)

{a(n) = (n+1) * (n+2) \ 6}; /* Michael Somos, Feb 11 2004 */

[binomial(n, 2) // 3 for n in range(2, 61)] # Zerinvary Lajos, Dec 01 2009

a(n) = (A000217(n+1) - A022003(n-1))/3;
a(n) = (A016754(n+1) - A010881(A016754(n+1)))/24;
a(n) = (A033996(n+1) - A010881(A033996(n+1)))/24.
Euler transform of length 3 sequence [2, 0, 1].
a(3*k-1) = k*(3*k + 1)/2;
a(3*k)   = 3*k*(k + 1)/2;
a(3*k+1) = (k + 1)*(3*k + 2)/2.
a(n) = floor( (n+1)*(n+2)/6 ) = floor( A000217(n+1)/3 ).
a(n+1) = a(n) + A008620(n) = A002264(n+3). - Reinhard Zumkeller, Aug 01 2002
From Michael Somos, Feb 11 2004: (Start)
G.f.: x / ((1-x)^2 * (1-x^3)).
a(n) = 1 + a(n-1) + a(n-3) - a(n-4).
a(-3-n) = a(n). (End)
a(n) = a(n-3) + n for n > 2; a(0)=0, a(1)=1, a(2)=2. - Paul Barry, Jul 14 2004
a(n) = binomial(n+3, 3)/(n+3) + cos(2*Pi*(n-1)/3)/9 + sqrt(3)sin(2*Pi*(n-1)/3)/9 - 1/9. - Paul Barry, Jan 01 2005
From Paul Barry, Apr 16 2005: (Start)
a(n) = Sum_{k=0..n} k*(cos(2*Pi*(n-k)/3 + Pi/3)/3 + sqrt(3)*sin(2*Pi*(n-k)/3 + Pi/3)/3 + 1/3).
a(n) = Sum_{k=0..floor(n/3)} n-3*k. (End)
For n > 1, a(n) = A000217(n) - a(n-1) - a(n-2); a(0)=0, a(1)=1.
G.f.: x/(1 + x + x^2)/(1 - x)^3. - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009
a(n) = (4 + 3*n^2 + 9*n)/18 + ((n mod 3) - ((n-1) mod 3))/9. - Klaus Brockhaus, Oct 01 2009
a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5), with n>4, a(0)=0, a(1)=1, a(2)=2, a(3)=3, a(4)=5. - Harvey P. Dale, Jul 25 2011
a(n) = A214734(n + 2, 1, 3). - Renzo Benedetti, Aug 27 2012
G.f.: x*G(0), where G(k) = 1 + x*(3*k+4)/(3*k + 2 - 3*x*(k+2)*(3*k+2)/(3*(1+x)*k + 6*x + 4 - x*(3*k+4)*(3*k+5)/(x*(3*k+5) + 3*(k+1)/G(k+1)))); (continued fraction). - Sergei N. Gladkovskii, Jun 10 2013
Empirical: a(n) = floor((n+3)/(e^(6/(n+3))-1)). - Richard R. Forberg, Jul 24 2013
a(n) = Sum_{i=0..n} floor((i+2)/3). - Bruno Berselli, Aug 29 2013
0 = a(n)*(a(n+2) + a(n+3)) + a(n+1)*(-2*a(n+2) - a(n+3) + a(n+4)) + a(n+2)*(a(n+2) - 2*a(n+3) + a(n+4)) for all n in Z. - Michael Somos, Jan 22 2014
a(n) = n/2 + floor(n^2/3 + 2/3)/2. - Bruno Berselli, Jan 23 2017
a(n) + a(n+1) = A000212(n+2). - R. J. Mathar, Jan 14 2021
Sum_{n>=1} 1/a(n) = 20/3 - 2*Pi/sqrt(3). - Amiram Eldar, Sep 27 2022
E.g.f.: (exp(x)*(4 + 12*x + 3*x^2) - 4*exp(-x/2)*cos(sqrt(3)*x/2))/18. - Stefano Spezia, Apr 05 2023

A007997 a(n) = ceiling((n-3)(n-4)/6).

Original entry on oeis.org

0, 0, 1, 1, 2, 4, 5, 7, 10, 12, 15, 19, 22, 26, 31, 35, 40, 46, 51, 57, 64, 70, 77, 85, 92, 100, 109, 117, 126, 136, 145, 155, 166, 176, 187, 199, 210, 222, 235, 247, 260, 274, 287, 301, 316, 330, 345, 361, 376, 392, 409, 425, 442, 460, 477, 495, 514, 532, 551, 571, 590, 610

Offset: 3

N. J. A. Sloane

Number of solutions to x+y+z=0 (mod m) with 0<=x<=y<=z

Nonorientable genus of complete graph on n nodes.

Also (with different offset) Molien series for alternating group A_3.

(1+x^3 ) / ((1-x)*(1-x^2)*(1-x^3)) is the Poincaré series [or Poincare series] (or Molien series) for H^*(S_6, F_2).

a(n+5) is the number of necklaces with 3 black beads and n white beads.

The g.f./x^5 is Z(C_3,x), the 3-variate cycle index polynomial for the cyclic group C_3, with substitution x[i]->1/(1-x^i), i=1,2,3. Therefore by Polya enumeration a(n+5) is the number of cyclically inequivalent 3-necklaces whose 3 beads are labeled with nonnegative integers such that the sum of labels is n, for n=0,1,2,... . See A102190 for Z(C_3,x). - Wolfdieter Lang, Feb 15 2005

a(n+1) is the number of pairs (x,y) with x and y in {0,...,n}, x = (y mod 3), and x+y < n. - Clark Kimberling, Jul 02 2012

From Gus Wiseman, Oct 17 2020: (Start)

Also the number of 3-part integer compositions of n - 2 that are either weakly increasing or strictly decreasing. For example, the a(5) = 1 through a(13) = 15 compositions are:

(111) (112) (113) (114) (115) (116) (117) (118) (119)

(122) (123) (124) (125) (126) (127) (128)

(222) (133) (134) (135) (136) (137)

(321) (223) (224) (144) (145) (146)

(421) (233) (225) (226) (155)

(431) (234) (235) (227)

(521) (333) (244) (236)

(432) (334) (245)

(531) (532) (335)

(621) (541) (344)

(631) (542)

(721) (632)

(641)

(731)

(821)

(End)

			For m=7 (n=12), the 12 solutions are xyz = 000 610 520 511 430 421 331 322 662 653 644 554.

A. Adem and R. J. Milgram, Cohomology of Finite Groups, Springer-Verlag, 2nd. ed., 2004, p. 204.
D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 105.
J. L. Gross and T. W. Tucker, Topological Graph Theory, Wiley, 1987; see \bar{I}(n) p. 221.
J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 740.
E. V. McLaughlin, Numbers of factorizations in non-unique factorial domains, Senior Thesis, Allegeny College, Meadville, PA, 2004.

T. D. Noe, Table of n, a(n) for n = 3..1000
Chwas Ahmed, Paul Martin, and Volodymyr Mazorchuk, On the number of principal ideals in d-tonal partition monoids, arXiv preprint arXiv:1503.06718 [math.CO], 2015-2019.
Alexander Taveira Blomenhofer, On Uniqueness of Power Sum Decomposition, SIAM J. Appl. Alg. Geom. (2025) Vol. 9, Iss. 1.
David Broadhurst and Xavier Roulleau, Number of partitions of modular integers, arXiv:2502.19523 [math.NT], 2025. See pp. 3, 19.
Magnus Rahbek Hansen, Invariant Theory and Graphs, Master's Thesis, Univ. Copenhagen (Denmark, 2023). See p. 38.
Mónica A. Reyes, Cristina Dalfó, Miguel Àngel Fiol, and Arnau Messegué, A general method to find the spectrum and eigenspaces of the k-token of a cycle, and 2-token through continuous fractions, arXiv:2403.20148 [math.CO], 2024. See p. 6.
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).
Index entries for Molien series.
Index entries for sequences related to necklaces.

Cf. A000031, A007998, A003050, A047996, A048259, A053618.
Apart from initial term, same as A058212.
Cf. A000212, A000217, A001840, A128422, A156040.
A001399(n-6)*2 = A069905(n-3)*2 = A211540(n-1)*2 counts the strict case.
A014311 intersected with A225620 U A333256 ranks these compositions.
A218004 counts these compositions of any length.
A000009 counts strictly decreasing compositions.
A000041 counts weakly increasing compositions.
A001523 counts unimodal compositions, with complement counted by A115981.
A007318 and A097805 count compositions by length.
A032020 counts strict compositions, ranked by A233564.
A333149 counts neither increasing nor decreasing strict compositions.
Cf. A014311, A332834, A333147, A337481, A337482-A337484.

a007997 n = ceiling $ (fromIntegral $ (n - 3) * (n - 4)) / 6
a007997_list = 0 : 0 : 1 : zipWith (+) a007997_list [1..]
-- Reinhard Zumkeller, Dec 18 2013

x^5*(1+x^3)/((1-x)*(1-x^2)*(1-x^3));
seq(ceil(binomial(n,2)/3), n=0..63); # Zerinvary Lajos, Jan 12 2009
a := n -> (n*(n-7)-2*([1,1,-1][n mod 3 +1]-7))/6;
seq(a(n), n=3..64); # Peter Luschny, Jan 13 2015

k = 3; Table[Apply[Plus, Map[EulerPhi[ # ]Binomial[n/#, k/# ] &, Divisors[GCD[n, k]]]]/n, {n, k, 30}] (* Robert A. Russell, Sep 27 2004 *)
Table[Ceiling[((n-3)(n-4))/6],{n,3,100}] (* or *) LinearRecurrence[ {2,-1,1,-2,1},{0,0,1,1,2},100] (* Harvey P. Dale, Jan 21 2014 *)

a(n)=(n^2-7*n+16)\6 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = a(n-3) + n - 2, a(0)=0, a(1)=0, a(2)=1 [Offset 0]. - Paul Barry, Jul 14 2004
G.f.: x^5*(1+x^3)/((1-x)*(1-x^2)*(1-x^3)) = x^5*(1-x+x^2)/((1-x)^2*(1-x^3)).
a(n+5) = Sum_{k=0..floor(n/2)} C(n-k,L(k/3)), where L(j/p) is the Legendre symbol of j and p. - Paul Barry, Mar 16 2006
a(3)=0, a(4)=0, a(5)=1, a(6)=1, a(7)=2, a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5). - Harvey P. Dale, Jan 21 2014
a(n) = (n^2 - 7*n + 14 - 2*(-1)^(2^(n + 1 - 3*floor((n+1)/3))))/6. - Luce ETIENNE, Dec 27 2014
a(n) = A001399(n-3) + A001399(n-6). Compare to A140106(n) = A001399(n-3) - A001399(n-6). - Gus Wiseman, Oct 17 2020
a(n) = (40 + 3*(n - 7)*n - 4*cos(2*n*Pi/3) - 4*sqrt(3)*sin(2*n*Pi/3))/18. - Stefano Spezia, Dec 14 2021
Sum_{n>=5} 1/a(n) = 6 - 2*Pi/sqrt(3) + 2*Pi*tanh(sqrt(5/3)*Pi/2)/sqrt(15). - Amiram Eldar, Oct 01 2022

A008810 a(n) = ceiling(n^2/3).

Original entry on oeis.org

0, 1, 2, 3, 6, 9, 12, 17, 22, 27, 34, 41, 48, 57, 66, 75, 86, 97, 108, 121, 134, 147, 162, 177, 192, 209, 226, 243, 262, 281, 300, 321, 342, 363, 386, 409, 432, 457, 482, 507, 534, 561, 588, 617, 646, 675, 706, 737, 768, 801, 834, 867, 902, 937, 972, 1009, 1046

Offset: 0

N. J. A. Sloane

a(n+1) is the number of 3-tuples (w,x,y) having all terms in {0,...,n} and 3*w = 2*x + y. - Clark Kimberling, Jun 04 2012

a(n) is also the number of L-shapes (3-cell polyominoes) packing into an n X n square. See illustration in links. - Kival Ngaokrajang, Nov 10 2013

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, number of red blocks in Fig 2.5.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
S. Lafortune, A. Ramani, B. Grammaticos, Y. Ohta and K.M. Tamizhmani, Blending two discrete integrability criteria: ..., arXiv:nlin/0104020 [nlin.SI], 2001.
Kival Ngaokrajang, Illustration of initial terms.
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Cf. A000290, A056105, A056109.

Cf. Expansions of the form (1+x^m)/((1-x)^2*(1-x^m)): A000290 (m=1), A000982 (m=2), this sequence (m=3), A008811 (m=4), A008812 (m=5), A008813 (m=6), A008814 (m=7), A008815 (m=8), A008816 (m=9), A008817 (m=10).

a008810 = ceiling . (/ 3) . fromInteger . a000290
a008810_list = [0,1,2,3,6] ++ zipWith5
               (\u v w x y -> 2 * u - v + w - 2 * x + y)
   (drop 4 a008810_list) (drop 3 a008810_list) (drop 2 a008810_list)
   (tail a008810_list) a008810_list
-- Reinhard Zumkeller, Dec 20 2012

[Ceiling(n^2/3): n in [0..60]]; // G. C. Greubel, Sep 12 2019

seq(ceil(n^2/3), n=0..60); # G. C. Greubel, Sep 12 2019

Ceiling[Range[0,60]^2/3] (* Vladimir Joseph Stephan Orlovsky, Mar 15 2011 *)
LinearRecurrence[{2,-1,1,-2,1},{0,1,2,3,6},60] (* Harvey P. Dale, Jun 20 2011 *)

a(n)=ceil(n^2/3) /* Michael Somos, Aug 03 2006 */

[ceil(n^2/3) for n in (0..60)] # G. C. Greubel, Sep 12 2019

a(-n) = a(n) = ceiling(n^2/3).
G.f.: x*(1 + x^3)/((1 - x)^2*(1 - x^3)) = x*(1 - x^6)/((1 - x)*(1 - x^3))^2.
From Michael Somos, Aug 03 2006: (Start)
Euler transform of length 6 sequence [ 2, 0, 2, 0, 0, -1].
a(3n-1) = A056105(n).
a(3n+1) = A056109(n). (End)
a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5) for n > 4. - Harvey P. Dale, Jun 20 2011
a(A008585(n)) = A033428(n). - Reinhard Zumkeller, Dec 20 2012
9*a(n) = 4 + 3*n^2 - 2*A099837(n+3). - R. J. Mathar, May 02 2013
a(n) = n^2 - 2*A000212(n). - Wesley Ivan Hurt, Jul 07 2013
Sum_{n>=1} 1/a(n) = Pi^2/18 + sqrt(2)*Pi*sinh(2*sqrt(2)*Pi/3)/(1+2*cosh(2*sqrt(2)*Pi/3)). - Amiram Eldar, Aug 13 2022
E.g.f.: (exp(x)*(4 + 3*x*(1 + x)) - 4*exp(-x/2)*cos(sqrt(3)*x/2))/9. - Stefano Spezia, Oct 28 2022

A016778 a(n) = (3*n+1)^2.

Original entry on oeis.org

1, 16, 49, 100, 169, 256, 361, 484, 625, 784, 961, 1156, 1369, 1600, 1849, 2116, 2401, 2704, 3025, 3364, 3721, 4096, 4489, 4900, 5329, 5776, 6241, 6724, 7225, 7744, 8281, 8836, 9409, 10000, 10609, 11236, 11881, 12544, 13225, 13924, 14641, 15376, 16129, 16900, 17689

Offset: 0

N. J. A. Sloane

From Paul Curtz, Mar 28 2019: (Start)
Sequence is a spoke of the hexagonal spiral built from the terms of A016777:
.
        \
       100--97--94--91
          \           \
          49--46--43  88
          / \       \   \
        52  16--13  40  85
        /   / \   \   \   \
      55  19   1  10  37  82
      /   /   /   /   /   /
    58  22   4---7  34  79
      \   \         /   /
      61  25--28--31  76
        \             /
        64--67--70--73
(End)

Shawn A. Broyles, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000212, A000290, A016777, A016779, A016780, A016781, A214550, A247792, A262178.

(3*Range[0,50]+1)^2 (* or *) LinearRecurrence[{3,-3,1},{1,16,49},50] (* Harvey P. Dale, Mar 03 2013 *)

A016778(n):=(3*n+1)^2$
makelist(A016778(n),n,0,20); /* Martin Ettl, Nov 12 2012 */

a(n)=(3*n+1)^2 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = a(n-1) + 3*(6*n-1); a(0)=1. - Vincenzo Librandi, Nov 20 2010
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=1, a(1)=16, a(2)=49. - Harvey P. Dale, Mar 03 2013
a(n) = A247792(n) + 6*n. - Miquel Cerda, Oct 23 2016
G.f.: (1 + 13*x + 4*x^2)/(1 - x)^3. - Ilya Gutkovskiy, Oct 23 2016
a(n) = A000212(3*n) + A000212(1+3*n) + A000212(2+3*n). - Paul Curtz, Mar 28 2019
From Amiram Eldar, Nov 12 2020: (Start)
Sum_{n>=0} 1/a(n) = A214550.
Sum_{n>=0} (-1)^n/a(n) = A262178. (End)
From Elmo R. Oliveira, May 29 2025: (Start)
E.g.f.: exp(x)*(1 + 15*x + 9*x^2).
a(n) = A000290(A016777(n)) = A016777(n)^2. (End)

A130519 a(n) = Sum_{k=0..n} floor(k/4). (Partial sums of A002265.)

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 3, 4, 6, 8, 10, 12, 15, 18, 21, 24, 28, 32, 36, 40, 45, 50, 55, 60, 66, 72, 78, 84, 91, 98, 105, 112, 120, 128, 136, 144, 153, 162, 171, 180, 190, 200, 210, 220, 231, 242, 253, 264, 276, 288, 300, 312, 325, 338, 351, 364, 378, 392, 406, 420, 435, 450

Offset: 0

Hieronymus Fischer, Jun 01 2007

Complementary to A130482 with respect to triangular numbers, in that A130482(n) + 4*a(n) = n(n+1)/2 = A000217(n).
Disregarding the first three 0's the resulting sequence a'(n) is the sum of the positive integers <= n that have the same residue modulo 4 as n. This is the additive counterpart of the quadruple factorial numbers. - Peter Luschny, Jul 06 2011
From Heinrich Ludwig, Dec 23 2017: (Start)
Column sums of (shift of rows = 4):
  1 2 3 4 5 6  7  8  9 10 11 12 13 14 ...
          1 2  3  4  5  6  7  8  9 10 ...
                     1  2  3  4  5  6 ...
                                 1  2 ...
  .......................................
  ---------------------------------------
  1 2 3 4 6 8 10 12 15 18 21 24 28 32 ...
  shift of rows = 1 see A000217
  shift of rows = 2 see A002620
  shift of rows = 3 see A001840
  shift of rows = 5 see A130520
(End)
Conjecture: a(n+2) is the maximum effective weight of a numerical semigroup S of genus n (see Nathan Pflueger). - Stefano Spezia, Jan 04 2019

			G.f. = x^4 + 2*x^5 + 3*x^6 + 4*x^7 + 6*x^8 + 8*x^9 + 10*x^10 + 12*x^11 + ...
[ n] a(n)
---------
[ 4] 1
[ 5] 2
[ 6] 3
[ 7] 4
[ 8] 1 + 5
[ 9] 2 + 6
[10] 3 + 7
[11] 4 + 8

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Bakir Farhi, On the Representation of the Natural Numbers as the Sum of Three Terms of the Sequence floor(n^2/a), Journal of Integer Sequences, Vol. 16 (2013), Article 13.6.4.
Darren Glass and Joshua Wagner, Arithmetical Structures on Paths With a Doubled Edge, arXiv:1903.01398 [math.CO], 2019.
Nathan Pflueger, On non-primitive Weierstrass points, Alg. Number Th. 12 (2018) 1923-1947 and arXiv:1608.0566 [math.AG], 2016.
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,1).

Cf. A000217, A001840, A002264, A002265, A002266, A002620, A004526, A010872, A010873, A010874, A130481, A130483, A130520.

Cf. A000290, A007590, A000212, A118015, A056827, A056834, A056838, A056865.

a:=List([0..65],n->Sum([0..n],k->Int(k/4)));; Print(a); # Muniru A Asiru, Jan 04 2019

[Round(n*(n-2)/8): n in [0..70]]; // Vincenzo Librandi, Jun 25 2011

quadsum := n -> add(k, k = select(k -> k mod 4 = n mod 4, [$1 .. n])):
A130519 := n ->`if`(n<3,0,quadsum(n-3)); seq(A130519(n),n=0..58); # Peter Luschny, Jul 06 2011

a[ n_] := Quotient[ (n - 1)^2, 8]; (* Michael Somos, Oct 14 2011 *)

makelist(floor((n-1)^2/8), n, 0, 70); /* Stefano Spezia, Jan 04 2019 */

{a(n) = (n - 1)^2 \ 8}; /* Michael Somos, Oct 14 2011 */

def A130519(n): return (n-1)**2>>3  # Chai Wah Wu, Jul 30 2022

G.f.: x^4/((1-x^4)*(1-x)^2) = x^4/((1+x)*(1+x^2)*(1-x)^3).
a(n) = +2*a(n-1) -1*a(n-2) +1*a(n-4) -2*a(n-5) +1*a(n-6).
a(n) = floor(n/4)*(n - 1 - 2*floor(n/4)) = A002265(n)*(n - 1 - 2*A002265(n)).
a(n) = (1/2)*A002265(n)*(n - 2 + A010873(n)).
a(n) = floor((n-1)^2/8). - Mitch Harris, Sep 08 2008
a(n) = round(n*(n-2)/8) = round((n^2-2*n-1)/8) = ceiling((n+1)*(n-3)/8). - Mircea Merca, Nov 28 2010
a(n) = A001972(n-4), n>3. - Franklin T. Adams-Watters, Jul 10 2009
a(n) = a(n-4)+n-3, n>3. - Mircea Merca, Nov 28 2010
Euler transform of length 4 sequence [ 2, 0, 0, 1]. - Michael Somos, Oct 14 2011
a(n) = a(2-n) for all n in Z. - Michael Somos, Oct 14 2011
a(n) = A214734(n, 1, 4). - Renzo Benedetti, Aug 27 2012
a(4n) = A000384(n), a(4n+1) = A001105(n), a(4n+2) = A014105(n), a(4n+3) = A046092(n). - Philippe Deléham, Mar 26 2013
a(n) = Sum_{i=1..ceiling(n/2)-1} (i mod 2) * (n - 2*i - 1). - Wesley Ivan Hurt, Jan 23 2014
a(n) = ( 2*n^2-4*n-1+(-1)^n+2*((-1)^((2*n-1+(-1)^n)/4)-(-1)^((6*n-1+(-1)^n)/4)) )/16 = ( 2*n*(n-2) - (1-(-1)^n)*(1-2*i^(n*(n-1))) )/16, where i=sqrt(-1). - Luce ETIENNE, Aug 29 2014
E.g.f.: (1/8)*((- 1 + x)*x*cosh(x) + 2*sin(x) + (- 1 - x + x^2)*sinh(x)). - Stefano Spezia, Jan 15 2019
a(n) = (A002620(n-1) - A011765(n+1)) / 2, for n > 0. - Yuchun Ji, Feb 05 2021
Sum_{n>=4} 1/a(n) = Pi^2/12 + 5/2. - Amiram Eldar, Aug 13 2022

Partially edited by R. J. Mathar, Jul 11 2009

A072704 Triangle of number of weakly unimodal partitions/compositions of n into exactly k terms.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 5, 4, 1, 1, 5, 8, 7, 5, 1, 1, 6, 12, 12, 9, 6, 1, 1, 7, 16, 20, 16, 11, 7, 1, 1, 8, 21, 30, 28, 20, 13, 8, 1, 1, 9, 27, 42, 45, 36, 24, 15, 9, 1, 1, 10, 33, 58, 68, 60, 44, 28, 17, 10, 1, 1, 11, 40, 77, 98, 95, 75, 52, 32, 19, 11, 1

Offset: 1

Henry Bottomley, Jul 04 2002

			Rows start:
01:  [1]
02:  [1, 1]
03:  [1, 2, 1]
04:  [1, 3, 3, 1]
05:  [1, 4, 5, 4, 1]
06:  [1, 5, 8, 7, 5, 1]
07:  [1, 6, 12, 12, 9, 6, 1]
08:  [1, 7, 16, 20, 16, 11, 7, 1]
09:  [1, 8, 21, 30, 28, 20, 13, 8, 1]
10:  [1, 9, 27, 42, 45, 36, 24, 15, 9, 1]
...
T(6,3)=8 since 6 can be written as 1+1+4, 1+2+3, 1+3+2, 1+4+1, 2+2+2, 2+3+1, 3+2+1, or 4+1+1 but not 2+1+3 or 3+1+2.

Alois P. Heinz, Rows n = 1..141, flattened
Henry Bottomley, Illustration of initial terms
Rigoberto Flórez, Leandro Junes, and José L. Ramírez, Enumerating several aspects of non-decreasing Dyck paths, Discrete Mathematics (2019) Vol. 342, Issue 11, 3079-3097. See page 3094, Table 4.
Eric Weisstein's World of Mathematics, Unimodal Sequence

Cf. A059623, A072705. Row sums are A001523. First column is A057427, second is A000027 offset, third appears to be A000212 offset, right hand columns include A000012, A000027, A005408 and A008574.
The case of partitions is A072233.
Dominates A332670 (the version for negated compositions).
The strict case is A072705.
The case of constant compositions is A113704.
Unimodal sequences covering an initial interval are A007052.
Partitions whose run-lengths are unimodal are A332280.
Cf. A107429, A115981, A227038, A328509, A332282, A332283, A332578, A332638, A332669, A332728.

b:= proc(n, i) option remember; local q; `if`(i>n, 0,
      `if`(irem(n, i, 'q')=0, x^q, 0) +expand(
      add(b(n-i*j, i+1)*(j+1)*x^j, j=0..n/i)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n, 1)):
seq(T(n), n=1..12);  # Alois P. Heinz, Mar 26 2014

b[n_, i_] := b[n, i] = If[i>n, 0, If[Mod[n, i ] == 0, x^Quotient[n, i], 0] + Expand[ Sum[b[n-i*j, i+1]*(j+1)*x^j, {j, 0, n/i}]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 1, n}]][b[n, 1]]; Table[T[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Feb 26 2015, after Alois P. Heinz *)
unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{k}],unimodQ]],{n,0,10},{k,0,n}] (* Gus Wiseman, Mar 06 2020 *)

\\ starting for n=0, with initial column 1, 0, 0, ...:
N=25;  x='x+O('x^N);
T=Vec(1 + sum(n=1, N, t*x^n / ( prod(k=1,n-1, (1 - t*x^k)^2 ) * (1 - t*x^n) ) ) )
for(r=1,#T, print(Vecrev(T[r])) ); \\ Joerg Arndt, Oct 01 2017

G.f. with initial column 1, 0, 0, ...: 1 + Sum_{n>=1} (t*x^n / ( ( Product_{k=1..n-1} (1 - t*x^k)^2 ) * (1 - t*x^n) ) ). - Joerg Arndt, Oct 01 2017

A056827 a(n) = floor(n^2/6).

Original entry on oeis.org

0, 0, 0, 1, 2, 4, 6, 8, 10, 13, 16, 20, 24, 28, 32, 37, 42, 48, 54, 60, 66, 73, 80, 88, 96, 104, 112, 121, 130, 140, 150, 160, 170, 181, 192, 204, 216, 228, 240, 253, 266, 280, 294, 308, 322, 337, 352, 368, 384, 400, 416, 433, 450, 468, 486, 504

Offset: 0

N. J. A. Sloane, Sep 02 2000

a(n-1) represents the floor of the area under the polygon connecting the lattice points (n, floor(n/3)) from 0..n, n>0 (see example). - Wesley Ivan Hurt, Jun 06 2014

			5|                                             .__.__.
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0|.__.__./_|__|__|__|__|__|__|__|__|__|__|__|__|__|__|_________________
  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17   .. n
  0  0  0  1  2  4  6  8 10 13 16 20 24 28 32 37 42 48   .. a(n)
     0  0  0  1  2  4  6  8 10 13 16 20 24 28 32 37 42   .. a(n-1) <--

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,1,-2,1).

Cf. A000290, A007590, A000212, A002620, A118015, A118013, A056834, A130519, A056838, A056865.

List([0..60], n-> Int(n^2/6) ); # G. C. Greubel, Jul 23 2019

[Floor(n^2/6): n in [0..60]]; // Vincenzo Librandi, May 08 2011

A056827:=n->floor(n^2/6); seq(A056827(k), k=0..60); # Wesley Ivan Hurt, Oct 29 2013

Floor[Range[0,60]^2/6] (* or *) LinearRecurrence[{2,-1,0,0,0,1,-2,1}, {0,0,0,1,2,4,6,8}, 60] (* Harvey P. Dale, Jun 06 2013 *)

n^2\6 \\ Charles R Greathouse IV, May 08 2011

[floor(n^2/6) for n in (0..60)] # G. C. Greubel, Jul 23 2019

From R. J. Mathar, Nov 22 2008: (Start)
G.f.: x^3*(1+x^2)/((1+x)*(1-x)^3*(1+x+x^2)*(1-x+x^2)).
a(n+1) - a(n) = A123919(n). (End)
a(n) = floor( (1/2) * Sum_{i=1..n+1} (ceiling(i/3) + floor(i/3) - 1) ). - Wesley Ivan Hurt, Jun 06 2014
Sum_{n>=3} 1/a(n) = 15/8 + Pi^2/36 - Pi/(4*sqrt(3)) + tan(Pi/(2*sqrt(3)))*Pi/(2*sqrt(3)). - Amiram Eldar, Aug 13 2022

A118015 a(n) = floor(n^2/5).

Original entry on oeis.org

0, 0, 0, 1, 3, 5, 7, 9, 12, 16, 20, 24, 28, 33, 39, 45, 51, 57, 64, 72, 80, 88, 96, 105, 115, 125, 135, 145, 156, 168, 180, 192, 204, 217, 231, 245, 259, 273, 288, 304, 320, 336, 352, 369, 387, 405, 423, 441, 460, 480, 500, 520, 540, 561, 583, 605, 627, 649, 672

Offset: 0

Reinhard Zumkeller, Apr 10 2006

It seems that for n >= 5, a(n) is the maximum number of non-overlapping 1 X 5 rectangles that can be packed into an n X n square. Rectangles can only be placed parallel to the sides of the square. Verified with Lobato's program. - Dmitry Kamenetsky, Aug 03 2009

Ismailescu & Lee prove that for n > 6, a(n) is composite. - Charles R Greathouse IV, Jan 10 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Dan Ismailescu and Yunkyu James Lee, Polynomially growing integer sequences all whose terms are composite, arXiv:2501.04851 [math.NT], 2025. See p. 1.
R. D. Lobato, Recursive partitioning approach for the Manufacturer's Pallet Loading Problem.
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,1,-2,1).

Cf. A000212, A000290, A007590, A002620, A056827, A118013, A279169.

Cf. A056834, A130519, A056838, A056865.

[ n^2 div 5: n in [0..58] ]; // Klaus Brockhaus, Nov 18 2008

Table[Floor[n^2/5], {n, 0, 60}] (* Bruno Berselli, Dec 12 2016 *)

a(n)=n^2\5 \\ Charles R Greathouse IV, Sep 24 2015

[int(n**2/5) for n in range(60)] # Bruno Berselli, Dec 12 2016

[floor(n^2/5) for n in range(60)] # Bruno Berselli, Dec 12 2016

G.f.: x^3*(1 + x)/((1 + x + x^2 + x^3 + x^4)*(1 - x)^3). - Klaus Brockhaus, Nov 18 2008
a(n) = A008732(n-4) + A008732(n-3). - R. J. Mathar, Nov 22 2008
a(5*m+r) = m*(5*m + 2*r) + a(r), with m >= 0 and 0 <= r < 5. Example: for m=4 and r=3, a(5*4+3) = a(23) = 4*(5*4 + 2*3) + a(3) = 104 + 1 = 105. - Bruno Berselli, Dec 12 2016
Sum_{n>=3} 1/a(n) = 25/16 + Pi^2/30 + sqrt(5-2*sqrt(5))*Pi/4. - Amiram Eldar, Aug 13 2022

Edited by Charles R Greathouse IV, Apr 20 2010

A156040 Number of compositions (ordered partitions) of n into 3 parts (some of which may be zero), where the first is at least as great as each of the others.

Original entry on oeis.org

1, 1, 3, 4, 6, 8, 11, 13, 17, 20, 24, 28, 33, 37, 43, 48, 54, 60, 67, 73, 81, 88, 96, 104, 113, 121, 131, 140, 150, 160, 171, 181, 193, 204, 216, 228, 241, 253, 267, 280, 294, 308, 323, 337, 353, 368, 384, 400, 417, 433, 451, 468, 486, 504, 523, 541, 561, 580, 600

Offset: 0

Jack W Grahl, Feb 02 2009, Feb 11 2009

For n = 1, 2 these are just the triangular numbers. a(n) is always at least 1/3 of the corresponding triangular number, since each partition of this type gives up to three ordered partitions with the same cyclical order.
An alternative definition, which avoids using parts of size 0: a(n) is the third diagonal of A184957. - N. J. A. Sloane, Feb 27 2011
Diagonal sums of the triangle formed by rows T(2, k) k = 0, 1, ..., 2m of ascending m-nomial triangles (see A004737):
1
1 2 1
1 2 3 2 1
1 2 3 4 3 2 1
1 2 3 4 5 4 3 2 1
1 2 3 4 5 6 5 4 3 2 1
- Bob Selcoe, Feb 07 2014
Arrange A004396 in rows successively shifted to the right two spaces and sum the columns:
1  1  2  3  3  4  5  5  6 ...
      1  1  2  3  3  4  5 ...
            1  1  2  3  3 ...
                  1  1  2 ...
                        1 ...
------------------------------
1  1  3  4  6  8 11 13 17 ... - L. Edson Jeffery, Jul 30 2014
a(n) is the dimension of three-dimensional (2n + 2)-homogeneous polynomial vector fields with full tetrahedral symmetry (for a given orthogonal representation), and which are solenoidal. - Giedrius Alkauskas, Sep 30 2017
Also the number of compositions of n + 3 into three parts, the first at least as great as each of the other two. Also the number of compositions of n + 4 into three parts, the first strictly greater than each of the other two. - Gus Wiseman, Oct 09 2020

			G.f. = 1 + x + 3*x^2 + 4*x^3 + 6*x^4 + 8*x^5 + 11*x^6 + 13*x^7 + 17*x^8 + 20*x^9 + ...
The a(4) = 6 compositions of 4 are: (4 0 0), (3 1 0), (3 0 1), (2 2 0), (2 1 1), (2 0 2).
From _Gus Wiseman_, Oct 05 2020: (Start)
The a(0) = 1 through a(7) = 13 triples of nonnegative integers summing to n where the first is at least as great as each of the other two are:
  (000)  (100)  (101)  (111)  (202)  (212)  (222)  (313)
                (110)  (201)  (211)  (221)  (303)  (322)
                (200)  (210)  (220)  (302)  (312)  (331)
                       (300)  (301)  (311)  (321)  (403)
                              (310)  (320)  (330)  (412)
                              (400)  (401)  (402)  (421)
                                     (410)  (411)  (430)
                                     (500)  (420)  (502)
                                            (501)  (511)
                                            (510)  (520)
                                            (600)  (601)
                                                   (610)
                                                   (700)
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Giedrius Alkauskas, Projective and polynomial superflows. I, arxiv.org/1601.06570 [math.AG] (2017), Section 4.3.
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

For compositions into 4 summands see A156039; also see A156041 and A156042.
Cf. A184957, A071619 (bisection).
A001399(n-2)*2 is the strict case.
A001840(n-2) is the version with opposite relations.
A001840(n-1) is the version with strict opposite relations.
A069905 is the case with strict relations.
A014311 ranks 3-part compositions, with strict case A337453.
A014612 ranks 3-part partitions, with strict case A007304.
Cf. A000217, A001523, A211540, A218004, A220377, A337483, A337484.

a:= proc(n) local m, r; m := iquo(n, 6, 'r'); (4 +6*m +2*r) *m + [1, 1, 3, 4, 6, 8][r+1] end: seq(a(n), n=0..60); # Alois P. Heinz, Jun 14 2009

nn = 58; CoefficientList[Series[x^3/(1 - x^2)^2/(1 - x^3) + 1/(1 - x^2)^2/(1 - x), {x, 0, nn}], x] (* Geoffrey Critzer, Jul 14 2013 *)
CoefficientList[Series[(1 + x^2)/((1 + x) * (1 + x + x^2) * (1 - x)^3), {x, 0, 58}], x] (* L. Edson Jeffery, Jul 29 2014 *)
LinearRecurrence[{1, 1, 0, -1, -1, 1}, {1, 1, 3, 4, 6, 8}, 60] (* Harvey P. Dale, May 28 2015 *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n+3,{3}],#[[1]]>=#[[2]]&&#[[1]]>=#[[3]]&]],{n,0,15}] (* Gus Wiseman, Oct 05 2020*)

{a(n) = n*(n+4)\6 + 1}; /* Michael Somos, Mar 26 2017 */

G.f.: (x^2+1) / (1-x-x^2+x^4+x^5-x^6). - Alois P. Heinz, Jun 14 2009
Slightly nicer g.f.: (1+x^2)/((1-x)*(1-x^2)*(1-x^3)). - N. J. A. Sloane, Apr 29 2011
a(n) = A007590(n+2) - A000212(n+2). - Richard R. Forberg, Dec 08 2013
a(2*n) = A071619(n+1). - L. Edson Jeffery, Jul 29 2014
a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6), with a(0) = 1, a(1) = 1, a(2) = 3, a(3) = 4, a(4) = 6, a(5) = 8. - Harvey P. Dale, May 28 2015
a(n) = (n^2 + 4*n + 3)/6 + IF(MOD(n, 2) = 0, 1/2) + IF(MOD(n, 3) = 1, -1/3). - Heinrich Ludwig, Mar 21 2017
a(n) = 1 + floor((n^2 + 4*n)/6). - Giovanni Resta, Mar 21 2017
Euler transform of length 4 sequence [1, 2, 1, -1]. - Michael Somos, Mar 26 2017
a(n) = a(-4 - n) for all n in Z. - Michael Somos, Mar 26 2017
0 = a(n)*(-1 + a(n) - 2*a(n+1) - 2*a(n+2) + 2*a(n+3)) + a(n+1)*(+1 + a(n+1) + 2*a(n+2) - 2*a(n+3)) + a(n+2)*(+1 + a(n+2) - 2*a(n+3)) + a(n+3)*(-1 + a(n+3)) for all n in Z. - Michael Somos, Mar 26 2017
a(n) = round((n+1)*(n+3)/6). - Bill McEachen, Feb 16 2021
Sum_{n>=0} 1/a(n) = 3/2 + Pi^2/36 + (tan(c1)-1)*c1 + 3*c2*sinh(c2)/(1+2*cosh(c2)), where c1 = Pi/(2*sqrt(3)) and c2 = Pi*sqrt(2)/3. - Amiram Eldar, Dec 10 2022
E.g.f.: ((16 + 15*x + 3*x^2)*cosh(x) + 2*exp(-x/2)*(cos(sqrt(3)*x/2) - sqrt(3)*sin(sqrt(3)*x/2)) + (7 + 15*x + 3*x^2)*sinh(x))/18. - Stefano Spezia, Apr 05 2023

More terms from Alois P. Heinz, Jun 14 2009

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cp's OEIS Frontend

A001840 Expansion of g.f. x/((1 - x)^2*(1 - x^3)).

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A337461 Number of pairwise coprime ordered triples of positive integers summing to n.

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A007997 a(n) = ceiling((n-3)(n-4)/6).

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A008810 a(n) = ceiling(n^2/3).

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A016778 a(n) = (3*n+1)^2.

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A130519 a(n) = Sum_{k=0..n} floor(k/4). (Partial sums of A002265.)

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GAP