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A063224
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Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 60 ).
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3
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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
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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FORMULA
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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) = 2/9*(6*(n+1) - 9 + 2*sqrt(3)*sin(2*(n + 1)*pi/3). - Eric W. Weisstein, Sep 10 2021
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MATHEMATICA
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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 *)
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PROG
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(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)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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