A063196 Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 7 ).

0, 1, 3, 3, 5, 5, 7, 7, 9, 9, 11, 11, 13, 13, 15, 15, 17, 17, 19, 19, 21, 21, 23, 23, 25, 25, 27, 27, 29, 29, 31, 31, 33, 33, 35, 35, 37, 37, 39, 39, 41, 41, 43, 43, 45, 45, 47, 47, 49, 49, 51, 51, 53, 53, 55, 55, 57, 57, 59, 59, 61, 61, 63, 63, 65, 65, 67, 67, 69, 69, 71, 71, 73, 73, 75, 75, 77, 77, 79, 79, 81, 81, 83

Offset: 1

Author

N. J. A. Sloane, Jul 10 2001

Keywords

• nonn
• easy

Comments

Also, for n>1, number of involutions (i.e. elements of order 2) in the dihedral group D_(n-1). - Lekraj Beedassy, Oct 22 2004

Also, the chromatic number of the n-th triangular graph; i.e., the chromatic index (edge chromatic number) of the n-th complete graph. - Danny Rorabaugh, Nov 26 2018

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• J. Sondow and E. W. Weisstein, MathWorld: Wallis Formula.
• William A. Stein, Dimensions of the spaces S_k^{new}(Gamma_0(N)).
• William A. Stein, The modular forms database.
• Eric Weisstein's World of Mathematics, Chromatic Number, Edge Chromatic Number, and Triangular Graph.
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Crossrefs

Cf. A109613.

Programs

Mathematica

CoefficientList[Series[-x (x^2 - 2 x - 1) / ((x - 1)^2 (x + 1)), {x, 0, 100}], x] (* Vincenzo Librandi, Nov 27 2018 *)
LinearRecurrence[{1,1,-1},{0,1,3,3},90] (* Harvey P. Dale, Sep 11 2024 *)

PARI

concat([0], Vec(-x^2*(x^2-2*x-1)/((x-1)^2*(x+1)) + O(x^100))) \\ Colin Barker, Sep 08 2013

Formula

For n > 1, a(n-1) = (2n + 1 + (-1)^n)/2 (odd numbers appearing twice). - Lekraj Beedassy, Oct 22 2004

For n > 1, a(n) = 2*n - a(n-1), (with a(1)=1). - Vincenzo Librandi, Dec 06 2010

From Colin Barker, Sep 08 2013: (Start)

a(n) = a(n-1) + a(n-2) - a(n-3) for n > 4.

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