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

%I A063224 #42 Feb 16 2025 08:32:45
%S A063224 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,
%T A063224 34,36,36,38,40,40,42,44,44,46,48,48,50,52,52,54,56,56,58,60,60,62,64,
%U A063224 64,66
%N A063224 Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 60 ).
%C A063224 Essentially the same as A063200, A273308.
%C A063224 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
%H A063224 William A. Stein, <a href="http://wstein.org/Tables/dimskg0new.gp">Dimensions of the spaces S_k^{new}(Gamma_0(N))</a>
%H A063224 William A. Stein, <a href="http://wstein.org/Tables/">The modular forms database</a>
%H A063224 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BishopGraph.html">Bishop Graph</a>
%H A063224 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TotalDominationNumber.html">Total Domination Number</a>
%H A063224 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,-1)
%F A063224 a(n) = 2 * A004523(n), n > 0. - _Wesley Ivan Hurt_, Sep 17 2013
%F A063224 G.f.: 2*x^2*(1+x) / ( (1+x+x^2)*(x-1)^2 ). - _R. J. Mathar_, Jul 15 2015
%F A063224 a(n) = (n-1) + floor((n-1)/3) + ((n-1) mod 3). - _Bruno Berselli_, Apr 04 2016
%F A063224 a(n) = 2*floor(2*n/3). - _Eric W. Weisstein_, Sep 10 2021
%F A063224 a(n) = a(n-1)+a(n-3)-a(n-4) for n > 4. - _Eric W. Weisstein_, Sep 10 2021
%F A063224 a(n) = 2/3*(ChebyshevU(n,-1/2)+2*n-1). - _Eric W. Weisstein_, Sep 10 2021
%F A063224 a(n) = 2/9*(6*(n+1) - 9 + 2*sqrt(3)*sin(2*(n + 1)*Pi/3)). - _Eric W. Weisstein_, Sep 10 2021
%t A063224 2 Floor[2 Range[20]/3] (* _Eric W. Weisstein_, Sep 10 2021 *)
%t A063224 LinearRecurrence[{1, 0, 1, -1}, {0, 2, 4, 4}, 2] (* _Eric W. Weisstein_, Sep 10 2021 *)
%t A063224 Table[2/3 (2 n - 1 + ChebyshevU[n, -1/2]), {n, 50}] (* _Eric W. Weisstein_, Sep 10 2021 *)
%t A063224 Table[2/9 (-9 + 6 (n + 1) + 2 Sqrt[3] Sin[2 (n + 1) Pi/3]), {n, 20}] (* _Eric W. Weisstein_, Sep 10 2021 *)
%t A063224 CoefficientList[Series[(2 x (1 + x))/((-1 + x)^2 (1 + x + x^2)), {x, 0, 20}], x] (* _Eric W. Weisstein_, Sep 10 2021 *)
%o A063224 (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
%o A063224 (Python)
%o A063224 def A063224(n): return n-1+sum(divmod(n-1,3)) # _Chai Wah Wu_, Jan 29 2023
%Y A063224 Cf. A063200, A273308.
%K A063224 nonn,easy
%O A063224 1,2
%A A063224 _N. J. A. Sloane_, Jul 10 2001