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A014679 G.f.: (1+x^3)^2/((1-x^2)*(1-x^3)*(1-x^4)).

%I A014679 #35 Aug 14 2022 17:11:16
%S A014679 1,0,1,3,2,3,6,6,7,10,11,13,16,17,20,24,25,28,33,35,38,43,46,50,55,58,
%T A014679 63,69,72,77,84,88,93,100,105,111,118,123,130,138,143,150,159,165,172,
%U A014679 181,188,196,205,212,221,231,238,247,258,266,275,286,295,305
%N A014679 G.f.: (1+x^3)^2/((1-x^2)*(1-x^3)*(1-x^4)).
%C A014679 Poincaré series [or Poincare series] (or Molien series) for mod 2 cohomology of M_12.
%D A014679 A. Adem and R. J. Milgram, Cohomology of Finite Groups, Springer-Verlag, 2nd. ed., 2004; p. 255, Theorem 3.20, where the series is given in the form GF_2 (see formula line).
%H A014679 T. D. Noe, <a href="/A014679/b014679.txt">Table of n, a(n) for n = 0..1000</a>
%H A014679 A. Adem, <a href="http://www.ams.org/notices/199707/adem.pdf">Recent developments in the cohomology of finite groups</a>, Notices Amer. Math. Soc., 44 (1997), 806-812.
%H A014679 Alejandro Adem; John Maginnis; James R. Milgram, <a href="http://dx.doi.org/10.1016/0021-8693(91)90285-G">The geometry and cohomology of the Mathieu group M_12</a>, J. Algebra 139 (1991), no. 1, 90-133.
%H A014679 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (2,-2,3,-3,2,-2,1).
%F A014679 Can also be written as GF_2 = (1 + x^2 + 3*x^3 + x^4 + 3*x^5 + 4*x^6 + 2*x^7 + 4*x^8 + 3*x^9 + x^10 + 3*x^11 + x^12 + x^14 ) / ( (1-x^4)*(1-x^6)*(1-x^7)).
%F A014679 G.f.: (1-x+x^2)^2/((1-x)^3*(1+x^2)(1+x+x^2)). a(n)=n^2/12+n/4+13/36-A057077(n)/4+4*A099837(n+3)/9. - _R. J. Mathar_, Jan 11 2009
%F A014679 a(0)=1, a(1)=0, a(2)=1, a(3)=3, a(4)=2, a(5)=3, a(6)=6, a(n)= 2*a(n-1)- 2*a(n-2)+3*a(n-3)-3*a(n-4)+2*a(n-5)-2*a(n-6)+a(n-7). - _Harvey P. Dale_, Apr 10 2012
%p A014679 (1+x^3)^2/((1-x^2)*(1-x^3)*(1-x^4));
%t A014679 CoefficientList[Series[(1+x^3)^2/((1-x^2)*(1-x^3)*(1-x^4)), {x,0,60}],x] (* _Harvey P. Dale_, Mar 17 2011 *)
%t A014679 LinearRecurrence[{2,-2,3,-3,2,-2,1},{1,0,1,3,2,3,6},60] (* _Harvey P. Dale_, Apr 10 2012 *)
%o A014679 (Magma) I:=[1,0,1,3,2,3,6]; [n le 7 select I[n] else 2*Self(n-1)- 2*Self(n-2)+3*Self(n-3)-3*Self(n-4)+2*Self(n-5)-2*Self(n-6)+Self(n-7): n in [1..60]]; // _Vincenzo Librandi_, Jul 19 2015
%o A014679 (PARI) a(n)=([0,1,0,0,0,0,0; 0,0,1,0,0,0,0; 0,0,0,1,0,0,0; 0,0,0,0,1,0,0; 0,0,0,0,0,1,0; 0,0,0,0,0,0,1; 1,-2,2,-3,3,-2,2]^n*[1;0;1;3;2;3;6])[1,1] \\ _Charles R Greathouse IV_, Feb 10 2017
%K A014679 nonn,easy,nice
%O A014679 0,4
%A A014679 _N. J. A. Sloane_