Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
A063224
Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 60 ).
3
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
William A. Stein, The modular forms database
Eric Weisstein's World of Mathematics, Bishop Graph
Eric Weisstein's World of Mathematics, Total Domination Number
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
Sequence in context: A230641 A373800 A063200 * A023847 A279667 A000061
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Jul 10 2001
STATUS
approved