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Expansion of (1-x+x^2+x^5)/((1-x)*(1-x^5)).
3

%I #18 Jan 23 2025 23:31:55

%S 1,0,1,1,1,3,2,3,3,3,5,4,5,5,5,7,6,7,7,7,9,8,9,9,9,11,10,11,11,11,13,

%T 12,13,13,13,15,14,15,15,15,17,16,17,17,17,19,18,19,19,19,21,20,21,21,

%U 21,23,22,23,23,23,25,24,25,25,25,27,26,27,27,27,29,28,29,29,29,31,30,31

%N Expansion of (1-x+x^2+x^5)/((1-x)*(1-x^5)).

%H G. C. Greubel, <a href="/A117451/b117451.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 (1,0,0,0,1,-1).

%F a(n) = a(n-1) + a(n-5) - a(n-6).

%F a(n) = -1 + ((5 + sqrt(5))/10)*cos(4*Pi*n/5) - sqrt((5 - sqrt(5))/250)*sin(4*Pi*n/5) + ((5-sqrt(5))/10)*cos(2*Pi*n/5) + sqrt((5+sqrt(5))/250)*sin(2*Pi*n/5) + (2*n + 5)/5.

%F a(n) = A117450(n) - A117450(n-1). - _G. C. Greubel_, Jun 03 2021

%t CoefficientList[Series[(1-x+x^2+x^5)/((1-x)(1-x^5)),{x,0,80}],x] (* or *) LinearRecurrence[{1,0,0,0,1,-1},{1,0,1,1,1,3},80] (* _Harvey P. Dale_, Jan 01 2016 *)

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 80); Coefficients(R!( (1-x+x^2+x^5)/((1-x)*(1-x^5)) )); // _G. C. Greubel_, Jun 03 2021

%o (Sage)

%o def A117451_list(prec):

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

%o return P( (1-x+x^2+x^5)/((1-x)*(1-x^5)) ).list()

%o A117451_list(80) # _G. C. Greubel_, Jun 03 2021

%Y Partial sums are A117450. Partial sums of A117452.

%K easy,nonn

%O 0,6

%A _Paul Barry_, Mar 16 2006