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

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

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

Comments

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)

Examples

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

References

• 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.

Links

• 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.

Crossrefs

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.

Programs

Haskell

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

Maple

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

Mathematica

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 *)

PARI

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

Formula

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

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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