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A117910 Expansion of (1 + x + x^2 + x^4)/((1-x^3)*(1-x^6)).

%I A117910 #26 Sep 08 2022 08:45:24
%S A117910 1,1,1,1,2,1,2,3,2,2,4,2,3,5,3,3,6,3,4,7,4,4,8,4,5,9,5,5,10,5,6,11,6,
%T A117910 6,12,6,7,13,7,7,14,7,8,15,8,8,16,8,9,17,9,9,18,9,10,19,10,10,20,10,
%U A117910 11,21,11,11,22,11,12,23,12,12,24,12,13,25,13,13,26,13,14,27,14
%N A117910 Expansion of (1 + x + x^2 + x^4)/((1-x^3)*(1-x^6)).
%C A117910 Diagonal sums of A117908.
%C A117910 Appears to be a permutation of floor((n+5)/5).
%H A117910 G. C. Greubel, <a href="/A117910/b117910.txt">Table of n, a(n) for n = 0..1000</a>
%H A117910 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,1,0,0,1,0,0,-1).
%F A117910 a(n) = a(n-3) + a(n-6) - a(n-9).
%F A117910 a(n) = Sum_{k=0..floor(n/2)} 0^abs(L(C(n-k,2)/3) - 2*L(C(k,2)/3)) where L(j/p) is the Legendre symbol of j and p.
%F A117910 From _G. C. Greubel_, Nov 18 2021: (Start)
%F A117910 a(n) = A152467(n+3) + A152467(n+6) if n == 1 (mod 3), otherwise A152467(n+6).
%F A117910 a(n) = A175676(n+2) if n == 1 (mod 3), otherwise A152467(n+6).
%F A117910 a(n) = A002264(n+3) if n == 1 (mod 3), otherwise A152467(n+6). (End)
%t A117910 CoefficientList[Series[(1+x+x^2+x^4)/((1-x^3)(1-x^6)),{x,0,100}],x] (* or *) LinearRecurrence[{0,0,1,0,0,1,0,0,-1},{1,1,1,1,2,1,2,3,2},100] (* _Harvey P. Dale_, Apr 10 2014 *)
%t A117910 Table[If[Mod[n,3]==1, Mod[Binomial[n+2,3], n+2], Floor[(n+6)/6]], {n, 0, 100}] (* _G. C. Greubel_, Nov 18 2021 *)
%o A117910 (Magma) R<x>:=PowerSeriesRing(Integers(), 100); Coefficients(R!( (1+x+x^2+x^4)/((1-x^3)*(1-x^6)) )); // _G. C. Greubel_, Oct 21 2021
%o A117910 (Sage)
%o A117910 def A117910_list(prec):
%o A117910     P.<x> = PowerSeriesRing(ZZ, prec)
%o A117910     return P( (1+x+x^2+x^4)/((1-x^3)*(1-x^6)) ).list()
%o A117910 A117910_list(100) # _G. C. Greubel_, Oct 21 2021
%Y A117910 Cf. A002264, A014125, A024164, A144677, A117908, A152467, A175676.
%K A117910 easy,nonn
%O A117910 0,5
%A A117910 _Paul Barry_, Apr 01 2006