OFFSET
3,5
COMMENTS
Number of solutions to x+y+z=0 (mod m) with 0<=x<=y<=z<m, where m = n-5.
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)
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
C. Ahmed, P. Martin, and V. Mazorchuk, On the number of principal ideals in d-tonal partition monoids, arXiv preprint arXiv:1503.06718 [math.CO], 2015-2019.
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).
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
EXAMPLE
For m=7 (n=12), the 12 solutions are xyz = 000 610 520 511 430 421 331 322 662 653 644 554.
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 *)
PROG
(Haskell)
a007997 n = ceiling $ (fromIntegral $ (n - 3) * (n - 4)) / 6
a007997_list = 0 : 0 : 1 : zipWith (+) a007997_list [1..]
-- Reinhard Zumkeller, Dec 18 2013
(PARI) a(n)=(n^2-7*n+16)\6 \\ Charles R Greathouse IV, Sep 24 2015
CROSSREFS
KEYWORD
nonn,nice,easy
AUTHOR
STATUS
approved