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A110514 Expansion of (1 - x + x^2 + x^3)/(1 - x^2 - x^4 + x^6).

%I A110514 #16 Aug 29 2017 22:51:14
%S A110514 1,-1,2,0,3,-1,4,0,5,-1,6,0,7,-1,8,0,9,-1,10,0,11,-1,12,0,13,-1,14,0,
%T A110514 15,-1,16,0,17,-1,18,0,19,-1,20,0,21,-1,22,0,23,-1,24,0,25,-1,26,0,27,
%U A110514 -1,28,0,29,-1,30,0,31,-1,32,0,33,-1,34,0,35,-1,36,0,37,-1,38,0,39,-1,40,0,41,-1,42,0,43,-1,44,0,45,-1,46,0,47,-1,48,0
%N A110514 Expansion of (1 - x + x^2 + x^3)/(1 - x^2 - x^4 + x^6).
%C A110514 Diagonal sums of A110515. Partial sums of A110516.
%H A110514 G. C. Greubel, <a href="/A110514/b110514.txt">Table of n, a(n) for n = 0..1000</a>
%H A110514 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,0,1,0,-1).
%F A110514 G.f.: (1 - x + x^2 + x^3)/((1 - x^2)^2(1 + x^2)).
%F A110514 a(n) = a(n-2) + a(n-4) - a(n-6).
%F A110514 a(n) = (n/2 + 1)*(1 + (-1)^n)/2 - (1 - (-1)^n)*(1 + (-1)^((n-1)/2))/4.
%F A110514 a(n) = (sin(Pi*n/2)*((-1)^n - 1) + (n+3)*(-1)^n + (n+1))/4.
%F A110514 a(n) = Sum_{k=0..floor(n/2)} Jacobi(2^(n-2k), 2(n-2k)+1) [conjecture].
%t A110514 Riffle[Range[50],{-1,0}] (* _Harvey P. Dale_, Dec 08 2011 *)
%o A110514 (PARI) x='x+O('x^50); Vec((1-x+x^2+x^3)/((1-x^2)^2(1+x^2))) \\ _G. C. Greubel_, Aug 29 2017
%Y A110514 Cf. A106249.
%K A110514 easy,sign
%O A110514 0,3
%A A110514 _Paul Barry_, Jul 24 2005