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%I #16 Sep 08 2022 08:44:36
%S 1,1,2,3,4,5,7,8,10,12,14,16,19,21,25,28,32,36,41,45,51,56,62,68,75,
%T 81,89,96,104,112,121,129,139,148,158,168,179,189,201,212,224,236,249,
%U 261,275,288,302,316
%N Expansion of (1+x^14)/((1-x)*(1-x^2)*(1-x^3)).
%H G. C. Greubel, <a href="/A008757/b008757.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-4)^2 + 287 + 9*(-1)^n + 8*(-1)^n*cos((n-2)*Pi/3) + 8*cos(2*n*Pi/3))/36 for n >= 9. - _G. C. Greubel_, Aug 04 2019
%t CoefficientList[Series[(1+x^14)/(1-x)/(1-x^2)/(1-x^3),{x,0,60}],x] (* or *) LinearRecurrence[{1,1,0,-1,-1,1},{1,1,2,3,4,5,7,8,10,12,14,16,19,21, 25},60] (* _Harvey P. Dale_, Jul 27 2017 *)
%t Join[{1,1,2,3,4,5,7,8,10}, Table[(6*(n-4)^2 +287 +9*(-1)^n +8*(-1)^n*Cos[(n-2)*Pi/3] +8*Cos[2*n*Pi/3])/36, {n,9,60}]] (* _G. C. Greubel_, Aug 04 2019 *)
%o (PARI) my(x='x+O('x^60)); Vec((1+x^14)/((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^14)/((1-x)*(1-x^2)*(1-x^3)) )); // _G. C. Greubel_, Aug 04 2019
%o (Sage) ((1+x^14)/((1-x)*(1-x^2)*(1-x^3))).series(x, 60).coefficients(x, sparse=False) # _G. C. Greubel_, Aug 04 2019
%o (GAP) a:=[12,14,16,19,21,25];; 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([1,1,2,3,4,5,7,8,10], a); # _G. C. Greubel_, Aug 04 2019
%K nonn
%O 0,3
%A _N. J. A. Sloane_