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%I #52 Jan 03 2023 09:24:54
%S 1,0,1,1,2,1,2,2,3,2,3,3,4,3,4,4,5,4,5,5,6,5,6,6,7,6,7,7,8,7,8,8,9,8,
%T 9,9,10,9,10,10,11,10,11,11,12,11,12,12,13,12,13,13,14,13,14,14,15,14,
%U 15,15,16,15,16,16,17
%N Expansion of g.f. (1 + x^3)/((1 - x^2)*(1 - x^4)) = (1 - x + x^2)/((1 + x)*(1 - x)^2*(1 + x^2)).
%C Molien series of 2-dimensional representation of group of order 16 over GF(3).
%D D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 107.
%H Vincenzo Librandi, <a href="/A008624/b008624.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Mo#Molien">Index entries for Molien series</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).
%F From _Reinhard Zumkeller_, Aug 05 2005: (Start)
%F a(n) = floor(n/4) + ((n mod 2 + 1 - floor((n mod 4)/3)) mod 2).
%F a(n) = A110654(A028242(n)). (End)
%F a(n) = (3 + 3*(-1)^n + (1-i)*(-i)^n + (1+i)*i^n + 2*n) / 8 where i = sqrt(-1). - _Colin Barker_, Oct 15 2015
%F a(n) = (2*n+3+2*cos(n*Pi/2)+3*cos(n*Pi)-2*sin(n*Pi/2))/8. - _Wesley Ivan Hurt_, Oct 01 2017
%F E.g.f.: (cos(x) + (3 + x)*cosh(x) - sin(x) + x*sinh(x))/4. - _Stefano Spezia_, Jan 03 2023
%p f := x -> (1+x^3)/((1-x^2)*(1-x^4)): seq(coeff(series(f(x), x, n+1), x, n), n=0..64);
%p a := n -> floor(n/4) + ((n mod 2 + 1 - floor((n mod 4)/3)) mod 2): seq(a(n), n=0..64); # _Johannes W. Meijer_, Oct 08 2013
%t CoefficientList[Series[(1 + x^3) / (1 - x^2) / (1 - x^4), {x, 0, 70}], x] (* _Vincenzo Librandi_, Aug 15 2013 *)
%o (PARI) a(n) = (3 + 3*(-1)^n + (1-I)*(-I)^n + (1+I)*I^n + 2*n) / 8 \\ _Colin Barker_, Oct 15 2015
%o (PARI) my(x='x+O('x^100)); Vec((1+x^3)/((1-x^2)*(1-x^4))) \\ _Altug Alkan_, Dec 24 2015
%Y Essentially the same as A059169.
%Y Cf. A028242, A110654, A110659.
%K nonn,easy
%O 0,5
%A _N. J. A. Sloane_
%E Replaced x^2 three times with x in the generating function (un-aerated). - _R. J. Mathar_, Oct 23 2008
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