A051275 Expansion of (1+x^2)/((1-x^2)*(1-x^3)).

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

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( 12 ).

Diagonal sums of A117567. - Paul Barry, Mar 29 2006

First differences of A156040. - Bob Selcoe, Feb 07 2014

Also first difference of diagonal sums of the triangle formed by rows T(2,k) k=0,1...,2m of ascending m-nomial triangles (see A004737). - Bob Selcoe, Feb 07 2014

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Luke James and Ben Salisbury, The weight function for monomial crystals of affine type, arXiv:1707.03159 [math.CO], 2017, p. 20 (sequence b_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).

Crossrefs

Cf. A051274, A117567, A156040.

Programs

Mathematica

CoefficientList[Series[(1+x^2)/((1-x^2)(1-x^3)),{x,0,100}],x] (* or *) LinearRecurrence[{0,1,1,0,-1},{1,0,2,1,2},100] (* Harvey P. Dale, Dec 10 2024 *)

PARI

Vec((1+x^2)/((1-x^2)*(1-x^3))+ O(x^80)) \\ Michel Marcus, Nov 26 2019

Formula

From Paul Barry, Mar 29 2006: (Start)

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

a(n) = cos(2*Pi*n/3 + Pi/3)/3 - sqrt(3)*sin(2*Pi*n/3 + Pi/3)/9 + (-1)^n/2 + (2n+3)/6;

a(n) = Sum_{k=0..floor(n/2)} F(L((n-2k+2)/3)) where L(j/p) is the Legendre symbol of j and p. (End)

a(n) = 2*floor(n/2) + floor((n+4)/3) - n. - Ridouane Oudra, Nov 26 2019