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Expansion of (1+x^12)/((1-x)*(1-x^2)*(1-x^3)).
1

%I #13 Sep 08 2022 08:44:36

%S 1,1,2,3,4,5,7,8,10,12,14,16,20,22,26,30,34,38,44,48,54,60,66,72,80,

%T 86,94,102,110,118,128,136,146,156,166,176,188,198,210,222,234,246,

%U 260,272,286,300,314,328

%N Expansion of (1+x^12)/((1-x)*(1-x^2)*(1-x^3)).

%H G. C. Greubel, <a href="/A008755/b008755.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,-1,-1,1).

%F a(n) = (6*n^2 -36*n +263 +9*(-1)^n +16*(-1)^n*cos(n*Pi/3))/36 for n >=7. - _G. C. Greubel_, Aug 04 2019

%t CoefficientList[Series[(1+x^12)/(1-x)/(1-x^2)/(1-x^3),{x,0,60}],x] (* _Harvey P. Dale_, Aug 27 2013 *)

%t Join[{1,1,2,3,4,5,7}, Table[(6*n^2 -36*n +263 +9*(-1)^n + 16*(-1)^n*Cos[n*Pi/3])/36, {n, 7, 60}]] (* _G. C. Greubel_, Aug 04 2019 *)

%o (PARI) my(x='x+O('x^60)); Vec((1+x^12)/((1-x)*(1-x^2)*(1-x^3))) \\ _G. C. Greubel_, Aug 04 2019

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 60); Coefficients(R!( (1+x^12)/((1-x)*(1-x^2)*(1-x^3)) )); // _G. C. Greubel_, Aug 04 2019

%o (Sage) ((1+x^12)/((1-x)*(1-x^2)*(1-x^3))).series(x, 60).coefficients(x, sparse=False) # _G. C. Greubel_, Aug 04 2019

%o (GAP) a:=[8,10,12,14,16,20];; for n in [7..60] do a[n]:=a[n-1]+a[n-2]-a[n-4]-a[n-5]+a[n-6]; od; Concatenation([1,1,2,3,4,5,7], a); # _G. C. Greubel_, Aug 04 2019

%K nonn

%O 0,3

%A _N. J. A. Sloane_