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

1,2

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

Table of n, a(n) for n=1..50.

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)

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

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 *)

Prog

(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

Crossrefs

Cf. A063200, A273308.

Sequence in context: A230641 A373800 A063200 * A023847 A279667 A000061

Adjacent sequences: A063221 A063222 A063223 * A063225 A063226 A063227

Keyword

nonn,easy

Author

N. J. A. Sloane, Jul 10 2001

Status

approved