A063093 Dimension of the space of weight 2n cusp forms for Gamma_0( 25 ).

0, 5, 9, 15, 19, 25, 29, 35, 39, 45, 49, 55, 59, 65, 69, 75, 79, 85, 89, 95, 99, 105, 109, 115, 119, 125, 129, 135, 139, 145, 149, 155, 159, 165, 169, 175, 179, 185, 189, 195, 199, 205, 209, 215, 219, 225, 229, 235, 239, 245

Offset: 1

Author

N. J. A. Sloane, Jul 08 2001

Keywords

• nonn
• easy

Comments

If b(n) is the sequence of integers congruent to {0,3} (mod 5) and c(n) is the sequence of integers congruent to {2,4}(mod 5). Then a(n) = b(n) + c(n). Equivalently a(n) = A047218(n+1) + A047211(n). - Anthony Hernandez, Aug 16 2016

Links

• William A. Stein, Dimensions of the spaces S_k(Gamma_0(N)).
• William A. Stein, The modular forms database.
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Crossrefs

Cf. A001622, A047211, A047218.

Programs

Mathematica

Rest@ CoefficientList[Series[x^2*(5 + 4 x + x^2)/((1 - x)^2*(1 + x)), {x, 0, 50}], x] (* Michael De Vlieger, Aug 26 2016 *)
LinearRecurrence[{1,1,-1},{0,5,9,15},50] (* Harvey P. Dale, Apr 09 2019 *)

Formula

a(n) = 10*n - a(n-1) - 16 for n>2, with a(1)=0, a(2)=5. - Vincenzo Librandi, Aug 07 2010

From Colin Barker, Sep 26 2012: (Start)

a(n) = ((-1)^n + 10*n - 11)/2 for n>1.

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

G.f.: x^2*(5+4*x+x^2)/((1-x)^2*(1+x)). (End)

Sum_{n>=2} (-1)^n/a(n) = sqrt(5+2*sqrt(5))*Pi/20 - 3*sqrt(5)*log(phi)/20 - log(5)/8, where phi is the golden ratio (A001622). - Amiram Eldar, Apr 15 2023