A063226 Dimension of the space of weight 2n cuspidal newforms for Gamma_0(63).

3, 7, 13, 17, 23, 27, 33, 37, 43, 47, 53, 57, 63, 67, 73, 77, 83, 87, 93, 97, 103, 107, 113, 117, 123, 127, 133, 137, 143, 147, 153, 157, 163, 167, 173, 177, 183, 187, 193, 197, 203, 207, 213, 217, 223, 227, 233, 237, 243, 247

Offset: 1

Author

N. J. A. Sloane, Jul 10 2001

Keywords

• nonn
• easy

Comments

Also, dimension of the space of weight 2n cuspidal newforms for Gamma_0(88). - N. J. A. Sloane, Nov 24 2016

First differences are 4,6,4,6,4,6.... Also values of k such that k^(10*n) mod 10 = 8*(n mod 2)+1. - Gary Detlefs, Jul 04 2014

In other words, numbers n such that n^(2+4*k) + 1 is divisible by 10, for k >= 0. - Altug Alkan, Mar 30 2016

The rational generating function, the periodic first differences and Greubel's closed form are an immediate consequence of the structure of formula given by [Martin]. - R. J. Mathar, Apr 09 2016

A quasipolynomial of order 2 and degree 1: a(n) = 5n - 3 if n is even and 5n - 2 if n is odd. - Charles R Greathouse IV, Nov 03 2021

Numbers that are congruent to {3, 7} mod 10. - Amiram Eldar, Nov 23 2024

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Greg Martin, Dimensions of the spaces of cusp forms and newforms on Gamma_0(N) and Gamma_1(N), J. Numb. Theory 112 (2005) 298-331, Theorem 1.
• 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 (1,1,-1).

Crossrefs

Cf. A017305 (bisection), A017353 (bisection), A019934, A182007.

Programs

Maple

# see A063195

Mathematica

Table[4 Floor[n/2] + 6 Floor[(n - 1)/2] + 3, {n, 50}] (* or *)
Table[SeriesCoefficient[3 x - x^2 (-7 - 6 x + 3 x^2)/((1 + x) (x - 1)^2), {x, 0, n}], {n, 50}] (* Michael De Vlieger, Mar 30 2016 *)
LinearRecurrence[{1, 1, -1}, {3, 7, 13}, 100] (* G. C. Greubel, Mar 30 2016 *)

PARI

my(x='x+O('x^99)); Vec(3*x-x^2*(-7-6*x+3*x^2)/((1+x)*(x-1)^2)) \\ Altug Alkan, Mar 31 2016

PARI

a(n)=5*n-3+n%2 \\ Charles R Greathouse IV, Mar 31 2016

Formula

a(n) = 4*floor(n/2) + 6*floor((n-1)/2) + 3. - Gary Detlefs, Jul 04 2014

G.f.: 3*x - x^2*(-7-6*x+3*x^2)/((1+x)*(x-1)^2). - R. J. Mathar, Jul 15 2015

From G. C. Greubel, Mar 30 2016: (Start)

a(n) = (1/2)*(10*n - 5 - (-1)^n).

E.g.f.: (5*x + 3)*cosh(x) + (5*x + 2)*sinh(x). (End)

Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(5-2*sqrt(5))*Pi/10. - Amiram Eldar, Sep 26 2022

From Amiram Eldar, Nov 23 2024: (Start)

Product_{n>=1} (1 - (-1)^n/a(n)) = 2*sin(Pi/5) (A182007).

Product_{n>=1} (1 + (-1)^n/a(n)) = tan(Pi/5) (A019934). (End)