A008612 Molien series of 2-dimensional representation of SL(2,3).

1, 0, 0, 1, 1, 0, 2, 1, 1, 2, 2, 1, 3, 2, 2, 3, 3, 2, 4, 3, 3, 4, 4, 3, 5, 4, 4, 5, 5, 4, 6, 5, 5, 6, 6, 5, 7, 6, 6, 7, 7, 6, 8, 7, 7, 8, 8, 7, 9, 8, 8, 9, 9, 8, 10, 9, 9, 10, 10, 9, 11, 10, 10, 11, 11, 10, 12, 11, 11, 12, 12, 11, 13, 12, 12, 13, 13, 12, 14

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0( 2 ).

References

• D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 100.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• William A. Stein, Dimensions of the spaces S_k^{new}(Gamma_0(N))
• William A. Stein, The modular forms database
• Index entries for Molien series
• Index entries for linear recurrences with constant coefficients, signature (0,1,1,0,-1).

Programs

Magma

[2*Floor(n/3)-n/2+(3+(-1)^n)/4: n in [0..100]]; // Vincenzo Librandi, Oct 23 2014

Maple

(1+x^12)/(1-x^6)/(1-x^8);seq(coeff(series(%,x,2*n+1),x,2*n), n=0..100);

Mathematica

CoefficientList[Series[(1-x^2+x^4)/((1-x)^2*(1+x)*(1+x+x^2)), {x, 0, 100}], x] (* Vincenzo Librandi, Oct 23 2014 *)

PARI

Vec((1-x^2+x^4)/((1-x)^2*(1+x)*(1+x+x^2)) + O(x^100)) \\ Colin Barker, Jan 07 2014

Sage

def A008612_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x^6)/((1-x^3)*(1-x^4)) ).list()
A008612_list(100) # G. C. Greubel, Feb 06 2020

Formula

From Colin Barker, Jan 07 2014: (Start)

a(n) = a(n-2) + a(n-3) - a(n-5).

G.f.: (1-x^2+x^4) / ((1-x)^2*(1+x)*(1+x+x^2)). (End)

a(n) ~ n/6 (first difference is 6-periodic). - Ralf Stephan, Apr 29 2014

a(n) = 2*floor(n/3) -n/2 +(3+(-1)^n)/4. - Tani Akinari, Oct 23 2014

12*a(n) = 1 +2*n +3*(-1)^n +8*A057078(n). - R. J. Mathar, Jan 14 2021