A063118 - OEIS

%I #50 Sep 26 2024 15:09:04

%S 2,17,31,47,61,77,91,107,121,137,151,167,181,197,211,227,241,257,271,

%T 287,301,317,331,347,361,377,391,407,421,437,451,467,481,497,511,527,

%U 541,557,571,587,601,617,631,647,661,677,691,707,721,737,751,767,781

%N Dimension of the space of weight 2n cusp forms for Gamma_0(50).

%C Appears to agree with the first 11-section of A186042 except for the first term of both sequences (verified up to a(10000)). - _Klaus Brockhaus_, Mar 10 2011

%H Klaus Brockhaus, <a href="/A063118/b063118.txt">Table of n, a(n) for n = 1..10000</a>

%H William A. Stein, <a href="http://wstein.org/Tables/dimskg0n.gp">Dimensions of the spaces S_k(Gamma_0(N))</a>.

%H William A. Stein, <a href="http://wstein.org/Tables/">The modular forms database</a>

%F From _Klaus Brockhaus_, Mar 10 2011: (Start)

%F G.f. (conjectured): x*(x^3 + 12*x^2 + 15*x + 2) / ((x - 1)^2*(x + 1)).

%F Recurrences (conjectured):

%F a(n) = a(n-1) + a(n-2) - a(n-3) for n > 4;

%F a(n) = a(n-2) + 30 for n > 3. (End)

%F Closed formula (conjectured): a(n) = (30*n+(-1)^n-27)/2 for n > 1. - _Bruno Berselli_, Mar 10 2011

%F Recurrence (conjectured): a(n) = 2*a(n-1) -a(n-2) +2*(-1)^n, n > 3. - _Vincenzo Librandi_, Mar 24 2011

%F Conjecture: a(n) = A007775(4*n - 3), n > 1. - _Bill McEachen_, May 15 2022

%e G.f. = 2*x + 17*x^2 + 31*x^3 + 47*x^4 + 61*x^5 + 77*x^6 + 91*x^7 + 107*x^8 + 121*x^9 + ...

%o (Magma) [ Dimension(CuspForms(Gamma0(50), 2*n)): n in [1..55] ]; // _Klaus Brockhaus_, Mar 10 2011

%o (Sage) def a(n) : return( len( CuspForms( Gamma0( 50), 2*n, prec=1) . basis())); # _Michael Somos_, May 29 2013

%Y Cf. A186042, A007775.

%K nonn

%O 1,1

%A _N. J. A. Sloane_, Jul 08 2001