cp's OEIS Frontend

A008612 Molien series of 2-dimensional representation of SL(2,3).

%I A008612 #35 Sep 08 2022 08:44:36
%S A008612 1,0,0,1,1,0,2,1,1,2,2,1,3,2,2,3,3,2,4,3,3,4,4,3,5,4,4,5,5,4,6,5,5,6,
%T A008612 6,5,7,6,6,7,7,6,8,7,7,8,8,7,9,8,8,9,9,8,10,9,9,10,10,9,11,10,10,11,
%U A008612 11,10,12,11,11,12,12,11,13,12,12,13,13,12,14
%N A008612 Molien series of 2-dimensional representation of SL(2,3).
%C A008612 Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0( 2 ).
%D A008612 D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 100.
%H A008612 Vincenzo Librandi, <a href="/A008612/b008612.txt">Table of n, a(n) for n = 0..1000</a>
%H A008612 William A. Stein, <a href="http://wstein.org/Tables/dimskg0new.gp">Dimensions of the spaces S_k^{new}(Gamma_0(N))</a>
%H A008612 William A. Stein, <a href="http://wstein.org/Tables/">The modular forms database</a>
%H A008612 <a href="/index/Mo#Molien">Index entries for Molien series</a>
%H A008612 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,1,0,-1).
%F A008612 From _Colin Barker_, Jan 07 2014: (Start)
%F A008612 a(n) = a(n-2) + a(n-3) - a(n-5).
%F A008612 G.f.: (1-x^2+x^4) / ((1-x)^2*(1+x)*(1+x+x^2)). (End)
%F A008612 a(n) ~ n/6 (first difference is 6-periodic). - _Ralf Stephan_, Apr 29 2014
%F A008612 a(n) = 2*floor(n/3) -n/2 +(3+(-1)^n)/4. - _Tani Akinari_, Oct 23 2014
%F A008612 12*a(n) = 1 +2*n +3*(-1)^n +8*A057078(n). - _R. J. Mathar_, Jan 14 2021
%p A008612 (1+x^12)/(1-x^6)/(1-x^8);seq(coeff(series(%,x,2*n+1),x,2*n), n=0..100);
%t A008612 CoefficientList[Series[(1-x^2+x^4)/((1-x)^2*(1+x)*(1+x+x^2)), {x, 0, 100}], x] (* _Vincenzo Librandi_, Oct 23 2014 *)
%o A008612 (PARI) Vec((1-x^2+x^4)/((1-x)^2*(1+x)*(1+x+x^2)) + O(x^100)) \\ _Colin Barker_, Jan 07 2014
%o A008612 (Magma) [2*Floor(n/3)-n/2+(3+(-1)^n)/4: n in [0..100]]; // _Vincenzo Librandi_, Oct 23 2014
%o A008612 (Sage)
%o A008612 def A008612_list(prec):
%o A008612     P.<x> = PowerSeriesRing(ZZ, prec)
%o A008612     return P( (1+x^6)/((1-x^3)*(1-x^4)) ).list()
%o A008612 A008612_list(100) # _G. C. Greubel_, Feb 06 2020
%K A008612 nonn,easy
%O A008612 0,7
%A A008612 _N. J. A. Sloane_