A063224 Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 60 ).

0, 2, 4, 4, 6, 8, 8, 10, 12, 12, 14, 16, 16, 18, 20, 20, 22, 24, 24, 26, 28, 28, 30, 32, 32, 34, 36, 36, 38, 40, 40, 42, 44, 44, 46, 48, 48, 50, 52, 52, 54, 56, 56, 58, 60, 60, 62, 64, 64, 66

Offset: 1

Author

N. J. A. Sloane, Jul 10 2001

Keywords

• nonn
• easy

Comments

Essentially the same as A063200, A273308.

Also the total domination number of the n X n bishop graph for n > 2. (Differs at the term a(2) since the 2 X 2 bishop graph has total domination number of 4.) - Eric W. Weisstein, Sep 10 2021

Links

• 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, Bishop Graph
• Eric Weisstein's World of Mathematics, Total Domination Number
• Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1)

Crossrefs

Cf. A063200, A273308.

Programs

Mathematica

2 Floor[2 Range[20]/3] (* Eric W. Weisstein, Sep 10 2021 *)
LinearRecurrence[{1, 0, 1, -1}, {0, 2, 4, 4}, 2] (* Eric W. Weisstein, Sep 10 2021 *)
Table[2/3 (2 n - 1 + ChebyshevU[n, -1/2]), {n, 50}] (* Eric W. Weisstein, Sep 10 2021 *)
Table[2/9 (-9 + 6 (n + 1) + 2 Sqrt[3] Sin[2 (n + 1) Pi/3]), {n, 20}] (* Eric W. Weisstein, Sep 10 2021 *)
CoefficientList[Series[(2 x (1 + x))/((-1 + x)^2 (1 + x + x^2)), {x, 0, 20}], x] (* Eric W. Weisstein, Sep 10 2021 *)

PARI

x='x+O('x^99); concat(0, Vec(2*x^2*(1+x)/((1+x+x^2)*(x-1)^2))) \\ Altug Alkan, Apr 04 2016

Python

def A063224(n): return n-1+sum(divmod(n-1,3)) # Chai Wah Wu, Jan 29 2023

Formula

a(n) = 2 * A004523(n), n > 0. - Wesley Ivan Hurt, Sep 17 2013

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

a(n) = (n-1) + floor((n-1)/3) + ((n-1) mod 3). - Bruno Berselli, Apr 04 2016

a(n) = 2*floor(2*n/3). - Eric W. Weisstein, Sep 10 2021

a(n) = a(n-1)+a(n-3)-a(n-4) for n > 4. - Eric W. Weisstein, Sep 10 2021

a(n) = 2/3*(ChebyshevU(n,-1/2)+2*n-1). - Eric W. Weisstein, Sep 10 2021

a(n) = 2/9*(6*(n+1) - 9 + 2*sqrt(3)*sin(2*(n + 1)*Pi/3)). - Eric W. Weisstein, Sep 10 2021