A051274 Expansion of (1+x^4)/((1-x^2)*(1-x^3)).

1, 0, 1, 1, 2, 1, 3, 2, 3, 3, 4, 3, 5, 4, 5, 5, 6, 5, 7, 6, 7, 7, 8, 7, 9, 8, 9, 9, 10, 9, 11, 10, 11, 11, 12, 11, 13, 12, 13, 13, 14, 13, 15, 14, 15, 15, 16, 15, 17, 16, 17, 17, 18, 17, 19, 18, 19, 19, 20, 19, 21, 20, 21, 21, 22, 21, 23, 22, 23, 23, 24, 23, 25

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( 3 ).

Links

• Luke James and Ben Salisbury, The weight function for monomial crystals of affine type, arXiv:1707.03159 [math.CO], 2017, p. 20 (sequence a_k).
• William A. Stein, Dimensions of the spaces S_k^{new}(Gamma_0(N))
• William A. Stein, The modular forms database
• Index entries for linear recurrences with constant coefficients, signature (0, 1, 1, 0, -1).

Programs

Mathematica

CoefficientList[Series[(1-x^8)/((1-x^2)*(1-x^3)*(1-x^4)),{x,0,90}],x] (* or *) LinearRecurrence[{0,1,1,0,-1},{1,0,1,1,2},90] (* Harvey P. Dale, Feb 20 2013 *)

PARI

a(n)=n\3+1-n%2 /* Michael Somos, Aug 26 2002 */

Formula

a(n) = 2*floor(n/2) + floor(n/3) - n + 1. Also a(0) = 1 and a(1) = 0, a(n) = a(n-2) + (a(n-1) reduced = (mod 2)). Again, a(0) = 1, a(1) = 0, a(n) = a(n-1) - 1 - (-1)^n - (a(n-2) mod 2). - Benoit Cloitre and Philippe Deléham, Jan 17 2004

a(n) = a(n-2) + a(n-3) - a(n-5). - Philippe Deléham, Sep 14 2006

Euler transform of length 8 sequence [ 0, 1, 1, 1, 0, 0, 0, -1]. - Michael Somos, Sep 26 2006

G.f.: (1-x^8)/((1-x^2)*(1-x^3)*(1-x^4)). a(n) = a(n-6) + 2. a(-1-n) = -a(n). - Michael Somos, Sep 26 2006