Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #13 Sep 08 2022 08:45:24
%S 1,1,3,3,3,5,6,4,8,8,6,10,11,7,13,13,9,15,16,10,18,18,12,20,21,13,23,
%T 23,15,25,26,16,28,28,18,30,31,19,33,33,21,35,36,22,38,38,24,40,41,25,
%U 43,43,27,45,46,28,48,48,30,50,51,31,53,53,33,55,56,34,58,58,36
%N Expansion of (1 + 2*x + 4*x^2 + 4*x^3 + 2*x^4)/((1+x)*(1-x^3)^2).
%C Diagonal sums of A117898.
%H G. C. Greubel, <a href="/A117900/b117900.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (-1,0,2,2,0,-1,-1).
%F a(n) = -a(n-1) + 2*a(n-3) + 2*a(n-4) - a(n-6) - a(n-7).
%F a(n) = Sum_{k=0..floor(n/2)} 2^abs(L(C(n-k,2)/3) - L(C(k,2)/3)), L(j/p) the Legendre symbol of j and p.
%t CoefficientList[Series[(1+2x+4x^2+4x^3+2x^4)/((1-x^3)(1+x-x^3-x^4)),{x,0,80}],x] (* or *) LinearRecurrence[{-1,0,2,2,0,-1,-1},{1,1,3,3,3,5,6},80] (* _Harvey P. Dale_, Mar 06 2018 *)
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 70); Coefficients(R!( (1+2*x+4*x^2+4*x^3+2*x^4)/((1+x)*(1-x^3)^2) )); // _G. C. Greubel_, Oct 01 2021
%o (Sage)
%o def A117899_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( (1+2*x+4*x^2+4*x^3+2*x^4)/((1+x)*(1-x^3)^2) ).list()
%o A117899_list(80) # _G. C. Greubel_, Oct 01 2021
%Y Cf. A117898.
%K easy,nonn
%O 0,3
%A _Paul Barry_, Apr 01 2006