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A008733 Molien series for 3-dimensional group [2+, n] = 2*(n/2).

%I A008733 #39 Sep 08 2022 08:44:36
%S A008733 1,0,2,1,4,2,6,4,9,6,12,9,16,12,20,16,25,20,30,25,36,30,42,36,49,42,
%T A008733 56,49,64,56,72,64,81,72,90,81,100,90,110,100,121,110,132,121,144,132,
%U A008733 156,144,169,156,182,169,196,182,210,196,225,210,240,225,256
%N A008733 Molien series for 3-dimensional group [2+, n] = 2*(n/2).
%H A008733 Vincenzo Librandi, <a href="/A008733/b008733.txt">Table of n, a(n) for n = 0..1000</a>
%H A008733 <a href="/index/Mo#Molien">Index entries for Molien series</a>
%H A008733 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1,1,-1,-1,1).
%F A008733 From _R. J. Mathar_, Nov 04 2008: (Start)
%F A008733 a(n) = A005232(n) - A005232(n-1).
%F A008733 G.f.: (1-x+x^2)/((1-x)^3*(1+x)^2*(1+x^2)). (End)
%F A008733 a(n) = floor((n^2 + 5*n + 13 + 3*(n+1)*(-1)^n)/16). - _Tani Akinari_, Aug 23 2013
%F A008733 a(n) = Sum_{i=1..floor((n+4)/2)} floor((i-(n mod 2))/2). - _Wesley Ivan Hurt_, Mar 31 2014
%F A008733 a(n) = (2*n^2+10*n+13+3*(2*n+5)*(-1)^n+4*(-1)^((6*n-1+(-1)^n)/4))/32. - _Luce ETIENNE_, Jun 09 2015
%t A008733 CoefficientList[Series[(1+x^3)/((1-x^2)^2*(1-x^4)), {x,0,70}], x] (* _Vincenzo Librandi_, Aug 24 2013 *)
%t A008733 LinearRecurrence[{1,1,-1,1,-1,-1,1},{1,0,2,1,4,2,6},70] (* _Harvey P. Dale_, Nov 23 2015 *)
%o A008733 (Magma) [Floor((n^2+5*n+13+3*(n+1)*(-1)^n)/16): n in [0..70]]; // _Vincenzo Librandi_, Aug 24 2013
%o A008733 (PARI) a(n)=((n^2+5*n+13+3*(n+1)*(-1)^n))\16 \\ _Charles R Greathouse IV_, Jun 11 2015
%o A008733 (Sage) [floor((n^2 + 5*n + 13 + 3*(n+1)*(-1)^n)/16) for n in (0..70)] # _G. C. Greubel_, Jul 30 2019
%o A008733 (GAP) List([0..70], n-> Int((n^2 + 5*n + 13 + 3*(n+1)*(-1)^n)/16)); # _G. C. Greubel_, Jul 30 2019
%K A008733 nonn,easy
%O A008733 0,3
%A A008733 _N. J. A. Sloane_