This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A051275 #39 Dec 10 2024 19:03:18
%S A051275 1,0,2,1,2,2,3,2,4,3,4,4,5,4,6,5,6,6,7,6,8,7,8,8,9,8,10,9,10,10,11,10,
%T A051275 12,11,12,12,13,12,14,13,14,14,15,14,16,15,16,16,17,16,18,17,18,18,19,
%U A051275 18,20,19,20,20,21,20,22,21,22,22,23,22,24,23,24,24
%N A051275 Expansion of (1+x^2)/((1-x^2)*(1-x^3)).
%C A051275 Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0( 12 ).
%C A051275 Diagonal sums of A117567. - _Paul Barry_, Mar 29 2006
%C A051275 First differences of A156040. - _Bob Selcoe_, Feb 07 2014
%C A051275 Also first difference of diagonal sums of the triangle formed by rows T(2,k) k=0,1...,2m of ascending m-nomial triangles (see A004737). - _Bob Selcoe_, Feb 07 2014
%H A051275 Harvey P. Dale, <a href="/A051275/b051275.txt">Table of n, a(n) for n = 0..1000</a>
%H A051275 Luke James and Ben Salisbury, <a href="https://arxiv.org/abs/1707.03159">The weight function for monomial crystals of affine type</a>, arXiv:1707.03159 [math.CO], 2017, p. 20 (sequence b_k).
%H A051275 William A. Stein, <a href="http://wstein.org/Tables/dimskg0new.gp">Dimensions of the spaces S_k^{new}(Gamma_0(N))</a>
%H A051275 William A. Stein, <a href="http://wstein.org/Tables/">The modular forms database</a>
%H A051275 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,1,0,-1).
%F A051275 From _Paul Barry_, Mar 29 2006: (Start)
%F A051275 a(n) = a(n-2) + a(n-3) - a(n-5);
%F A051275 a(n) = cos(2*Pi*n/3 + Pi/3)/3 - sqrt(3)*sin(2*Pi*n/3 + Pi/3)/9 + (-1)^n/2 + (2n+3)/6;
%F A051275 a(n) = Sum_{k=0..floor(n/2)} F(L((n-2k+2)/3)) where L(j/p) is the Legendre symbol of j and p. (End)
%F A051275 a(n) = 2*floor(n/2) + floor((n+4)/3) - n. - _Ridouane Oudra_, Nov 26 2019
%t A051275 CoefficientList[Series[(1+x^2)/((1-x^2)(1-x^3)),{x,0,100}],x] (* or *) LinearRecurrence[{0,1,1,0,-1},{1,0,2,1,2},100] (* _Harvey P. Dale_, Dec 10 2024 *)
%o A051275 (PARI) Vec((1+x^2)/((1-x^2)*(1-x^3))+ O(x^80)) \\ _Michel Marcus_, Nov 26 2019
%Y A051275 Cf. A051274, A117567, A156040.
%K A051275 nonn,easy
%O A051275 0,3
%A A051275 _N. J. A. Sloane_