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%I #68 Sep 08 2022 08:46:05
%S 1,-1,1,-2,4,-5,9,-15,23,-38,62,-99,161,-261,421,-682,1104,-1785,2889,
%T -4675,7563,-12238,19802,-32039,51841,-83881,135721,-219602,355324,
%U -574925,930249,-1505175,2435423,-3940598,6376022,-10316619,16692641,-27009261,43701901,-70711162,114413064,-185124225
%N Expansion of (1-x+x^3)/(1-x^2+2*x^3-x^4).
%C a(n) and its differences:
%C . 1, -1, 1, -2, 4, -5, 9, -15, 23, -38, ...
%C . -2, 2, -3, 6, -9, 14, -24, 38, -61, 100, ...
%C . 4, -5, 9, -15, 23, -38, 62, -99, 161, -261, ...
%C . -9, 14, -24, 38, -61, 100, -161, 260, -422, 682, ...
%C . 23, -38, 62, -99, 161, -261, 421, -682, 1104, -1785, ...
%C . -61, 100, -161, 260, -422, 682, -1103, 1786, -2889, 4674, ...
%C . 161, -261, 421, -682, 1104, -1785, 2889, -4675, 7563, -12238, ...
%C The third row is the first shifted .
%C The main diagonal is A001077(n). The fourth is -A001077(n+1). By "shifted" antidiagonals there are one 1, two 2's (-2 of the first row and 2), generally A001651(n) (-1)^n *A001077(n).
%C a(n+1)/a(n) tends to A001622 (the golden ratio) as n -> infinity.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,-2,1).
%F a(0)=1, a(1)=-1; for n>1, a(n) = a(n-2) - a(n-1) + A010892(n+2).
%F a(n) = a(n-2) -2*a(n-3) +a(n-4).
%F a(n) = A226956(-n).
%F a(n+1) = A039834(n) - (-1)^n*A094686(n).
%F a(n+6) - a(n) = 2*(-1)^n* A000032(n+3).
%F a(2n+1) = -A226956(2n+1).
%F G.f. ( -1+x-x^3 ) / ( (x^2-x-1)*(1-x+x^2) ). - _R. J. Mathar_, Jun 29 2013
%F 2*a(n) = A010892(n+2)+A061084(n+1). - _R. J. Mathar_, Jun 29 2013
%t a[0] = 1; a[1] = -1; a[n_] := a[n] = a[n-2] - a[n-1] - {-1, 0, 1, 1, 0, -1}[[Mod[n+1, 6] + 1]]; Table[a[n], {n, 0, 41}] (* _Jean-François Alcover_, Jul 04 2013 *)
%o (Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x+x^3)/(1-x^2+2*x^3-x^4))); // _Bruno Berselli_, Jul 04 2013
%K sign,easy
%O 0,4
%A _Paul Curtz_, Jun 28 2013
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