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A051274 Expansion of (1+x^4)/((1-x^2)*(1-x^3)). 5

%I #29 Aug 22 2019 12:18:12

%S 1,0,1,1,2,1,3,2,3,3,4,3,5,4,5,5,6,5,7,6,7,7,8,7,9,8,9,9,10,9,11,10,

%T 11,11,12,11,13,12,13,13,14,13,15,14,15,15,16,15,17,16,17,17,18,17,19,

%U 18,19,19,20,19,21,20,21,21,22,21,23,22,23,23,24,23,25

%N Expansion of (1+x^4)/((1-x^2)*(1-x^3)).

%C Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0( 3 ).

%H 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 a_k).

%H William A. Stein, <a href="http://wstein.org/Tables/dimskg0new.gp">Dimensions of the spaces S_k^{new}(Gamma_0(N))</a>

%H William A. Stein, <a href="http://wstein.org/Tables/">The modular forms database</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (0, 1, 1, 0, -1).

%F a(n) = 2*floor(n/2) + floor(n/3) - n + 1. Also a(0) = 1 and a(1) = 0, a(n) = a(n-2) + (a(n-1) reduced = (mod 2)). Again, a(0) = 1, a(1) = 0, a(n) = a(n-1) - 1 - (-1)^n - (a(n-2) mod 2). - _Benoit Cloitre_ and _Philippe Deléham_, Jan 17 2004

%F a(n) = a(n-2) + a(n-3) - a(n-5). - _Philippe Deléham_, Sep 14 2006

%F Euler transform of length 8 sequence [ 0, 1, 1, 1, 0, 0, 0, -1]. - _Michael Somos_, Sep 26 2006

%F G.f.: (1-x^8)/((1-x^2)*(1-x^3)*(1-x^4)). a(n) = a(n-6) + 2. a(-1-n) = -a(n). - _Michael Somos_, Sep 26 2006

%t CoefficientList[Series[(1-x^8)/((1-x^2)*(1-x^3)*(1-x^4)),{x,0,90}],x] (* or *) LinearRecurrence[{0,1,1,0,-1},{1,0,1,1,2},90] (* _Harvey P. Dale_, Feb 20 2013 *)

%o (PARI) a(n)=n\3+1-n%2 /* _Michael Somos_, Aug 26 2002 */

%K nonn,easy

%O 0,5

%A _N. J. A. Sloane_

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