%I #15 Sep 22 2025 16:00:21

%S 1,1,2,3,4,5,7,8,10,12,14,16,19,21,24,28,31,35,40,44,49,55,60,66,73,

%T 79,86,94,101,109,118,126,135,145,154,164,175,185,196,208,219,231,244,

%U 256,269,283,296,310

%N Expansion of (1+x^15)/((1-x)*(1-x^2)*(1-x^3)).

%H G. C. Greubel, <a href="/A008758/b008758.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,1,-2,1).

%F a(n) = (6*n^2 -54*n +452 + 8*(-1)^n*cos(n*Pi/3) + 8*cos(2*n*Pi/3))/36 for n>9. - _G. C. Greubel_, Aug 09 2019

%p seq(coeff(series((1+x^15)/((1-x)*(1-x^2)*(1-x^3)), x, n+1), x, n), n = 1 .. 60); # _G. C. Greubel_, Aug 09 2019

%t CoefficientList[Series[(1+x^15)/(1-x)/(1-x^2)/(1-x^3),{x,0,60}],x] (* _Harvey P. Dale_, Dec 28 2013 *)

%t Join[{1,1,2,3,4,5,7,8,10,12}, Table[(6*n^2 -54*n +452 + 8*(-1)^n*Cos[n*Pi/3] + 8*Cos[2*n*Pi/3])/36, {n,10,60}]] (* _G. C. Greubel_, Aug 09 2019 *)

%o (PARI) my(x='x+O('x^60)); Vec((1+x^15)/((1-x)*(1-x^2)*(1-x^3))) \\ _G. C. Greubel_, Aug 09 2019

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 60); Coefficients(R!( (1+x^15)/((1-x)*(1-x^2)*(1-x^3)) )); // _G. C. Greubel_, Aug 09 2019

%o (SageMath)

%o def A008758_list(prec):

%o P.<x> = PowerSeriesRing(ZZ, prec)

%o return P( (1+x^15)/((1-x)*(1-x^2)*(1-x^3)) ).list()

%o A008758_list(60) # _G. C. Greubel_, Aug 09 2019

%o (GAP) a:=[14,16,19,21,24];; for n in [6..30] do a[n]:=2*a[n-1]-a[n-2] +a[n-3]-2*a[n-4]+a[n-5]; od; Concatenation([1,1,2,3,4,5,7,8,10,12], a); # _G. C. Greubel_, Aug 09 2019

%K nonn

%O 0,3

%A _N. J. A. Sloane_