3,5

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 Poincare series (or Molien series) for H^*(S_6, F_2).

a(n-3)=number of necklaces with 3 black beads and n-3 white beads [I changed the offset for this sequence, so this statement will need to be adjusted]

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). W. Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Feb 15 2005.

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

Index entries for Molien series

Index entries for sequences related to necklaces

a(n)=a(n-3)+n-2, a(0)=0, a(1)=0, a(2)=1 [Offset 0]. - Paul Barry (pbarry(AT)wit.ie), 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 (pbarry(AT)wit.ie), Mar 16 2006

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

x^5*(1+x^3)/((1-x)*(1-x^2)*(1-x^3));

k = 3; Table[Apply[Plus, Map[EulerPhi[ # ]Binomial[n/#, k/# ] &, Divisors[GCD[n, k]]]]/n, {n, k, 30}] - Robert A. Russell (russell(AT)post.harvard.edu), Sep 27 2004

Cf. A000031, A007998, A003050, A047996, A048259, A053618.

Apart from initial term, same as A058212..

Sequence in context: A005653 A092311 A058212 this_sequence A123120 A036559 A083022

Adjacent sequences: A007994 A007995 A007996 this_sequence A007998 A007999 A008000

nonn,nice,easy

njas