A063289 Dimension of the space of weight n cuspidal newforms for Gamma_1( 16 ).

-1, 2, 7, 11, 16, 20, 25, 29, 34, 38, 43, 47, 52, 56, 61, 65, 70, 74, 79, 83, 88, 92, 97, 101, 106, 110, 115, 119, 124, 128, 133, 137, 142, 146, 151, 155, 160, 164, 169, 173, 178, 182, 187, 191, 196, 200, 205, 209, 214, 218, 223, 227, 232, 236

Offset: 2

Author

N. J. A. Sloane, Jul 14 2001

Keywords

• sign
• easy

Comments

It appears that for n > 2 a(n) = floor((9n-22)/2). - Gary Detlefs, Mar 02 2010

Links

• William A. Stein, Dimensions of the spaces S_k^{new}(Gamma_1(N)).
• William A. Stein, The modular forms database.
• Index entries for linear recurrences with constant coefficients, signature (1, 1, -1).

Crossrefs

Cf. A063232, A063233, A017185 (bisection), A130880, A332438.

Programs

Mathematica

Join[{-1}, Table[9*n/2 + (-1)^n/4 - 45/4, {n, 3, 60}]] (* Amiram Eldar, Jan 12 2024 *)

Formula

a(n) = 9*n/2 + (-1)^n/4 - 45/4 for n >= 3, with first differences in A010710. - R. J. Mathar, Dec 06 2010

From M. F. Hasler, Mar 05 2012: (Start)

G.f.: x^2*(-1 + 3*x + 6*x^2 + x^3)/(1 - x - x^2 + x^3).

a(n+2) = a(n)+9 (n>2), a(2n+1) = a(2n)+4 (n>1), a(2n) = a(2n-1)+5 (n>1). (End)

Sum_{n>=3} (-1)^(n+1)/a(n) = cot(2*Pi/9)*Pi/9. - Amiram Eldar, Jan 12 2024

From Amiram Eldar, Nov 22 2024: (Start)

Product_{n>=3} (1 - (-1)^n/a(n)) = 2*sin(Pi/18) + 1 (= A130880 + 1).

Product_{n>=3} (1 + (-1)^n/a(n)) = (1/2) * sec(Pi/9) (= A332438 - 3). (End)