A061866 a(n) is the number of solutions to x+y+z = 0 mod 3, where 1 <= x < y < z <= n.

0, 0, 0, 1, 2, 4, 8, 13, 20, 30, 42, 57, 76, 98, 124, 155, 190, 230, 276, 327, 384, 448, 518, 595, 680, 772, 872, 981, 1098, 1224, 1360, 1505, 1660, 1826, 2002, 2189, 2388, 2598, 2820, 3055, 3302, 3562, 3836, 4123, 4424, 4740, 5070, 5415, 5776, 6152, 6544, 6953, 7378, 7820

Offset: 0

Author

Antti Karttunen, May 11 2001

Keywords

• nonn
• easy

Comments

(1+x)*(1+x^2)*(1+x^3) / ( (1-x)*(1-x^2)*(1-x^3)*(1-x^4)) is the Poincaré series [or Poincare series] (or Molien series) for H^*(O_4(q); F_2).

References

• A. Adem and R. J. Milgram, Cohomology of Finite Groups, Springer-Verlag, 2nd. ed., 2004; p. 233.

Links

• D. Z. Dokovic, B. H. Smith, Quaternionic matrices: Unitary similarity, simultaneous triangularization and some trace identities, Lin. Alg. Applic. 428 (4) (2008) 890-910
• Index entries for linear recurrences with constant coefficients, signature (3,-3,2,-3,3,-1).

Crossrefs

The third diagonal of A061865.

Programs

Mathematica

LinearRecurrence[{3,-3,2,-3,3,-1},{0,0,0,1,2,4},60] (* Harvey P. Dale, Nov 22 2014 *)

Formula

G.f.: x^3*(1+x)*(1+x^2)*(1+x^3) / ( (1-x)*(1-x^2)*(1-x^3)*(1-x^4)). - N. J. A. Sloane, Mar 17 2004

a(n) = (binomial(n,3)+2*floor(n/3))/3. - Claude Morin, Mar 06 2012

G.f.: x^3*(1-x+x^2) / ( (1+x+x^2)*(x-1)^4 ). - R. J. Mathar, Dec 18 2014

a(n+5) = A014125(n)-A014125(n+1)+A014125(n+2). - R. J. Mathar, Mar 11 2019