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%I #42 May 10 2019 22:45:29
%S 1,-1,5,-10,30,-74,199,-515,1355,-3540,9276,-24276,63565,-166405,
%T 435665,-1140574,2986074,-7817630,20466835,-53582855,140281751,
%U -367262376,961505400,-2517253800,6590256025,-17253514249,45170286749,-118257345970,309601751190,-810547907570
%N Expansion of (1-x)^(-1)/(1 + 2*x - 2*x^2 - x^3).
%H G. C. Greubel, <a href="/A077916/b077916.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (-1,4,-1,-1).
%F a(n-1) = Sum_{i=1..n} (-1)^(i+1)*Fibonacci(i)*Fibonacci(i+1), n >= 1. - _Alexander Adamchuk_, Jun 16 2006
%F From _R. J. Mathar_, Mar 14 2011: (Start)
%F G.f.: 1/((1-x)^2*(1+3*x+x^2)).
%F a(n) = ((-1)^n*A001906(n+2)+n+2)/5. (End)
%F O.g.f.: exp( Sum_{n >= 1} Lucas(n)^2*(-x)^n/n ) = 1 - x + 5*x^2 - 10*x^3 + .... Cf. A203803. See also A207969 and A207970. - _Peter Bala_, Apr 03 2014
%F From _Vladimir Reshetnikov_, Oct 28 2015: (Start)
%F Recurrence (5-term): a(0) = 1, a(1) = -1, a(2) = 5, a(3) = -10, a(n) = -a(n-1) + 4*a(n-2) - a(n-3) - a(n-4).
%F Recurrence (4-term): a(0) = 1, a(1) = -1, a(2) = 5, n*a(n) = (1-2*n)*a(n-1) + (3*n+3)*a(n-2) + (n+1)*a(n-3).
%F (End)
%F a(n) = (F(2n+2)+n+1)/5 if n is odd and a(n)= -(F(2n+2)-n-1)/5 if n is even, where F(n) = Fibonacci numbers (A000045). - _Rigoberto Florez_, May 09 2019
%t a[0] = 1; a[1] = -1; a[2] = 5; a[3] = -10; a[n_] := a[n] = -a[n-1] + 4 a[n-2] - a[n-3] - a[n-4]; Table[a[n], {n, 0, 20}] (* _Vladimir Reshetnikov_, Oct 28 2015 *)
%t CoefficientList[Series[(1 - x)^(-1)/(1 + 2*x - 2*x^2 - x^3), {x, 0, 50}], x] (* _G. C. Greubel_, Dec 25 2017 *)
%t Table[If[OddQ[n], (Fibonacci[2n+2]+n+1)/5, -(Fibonacci[2n+2]-n-1)/5], {n,1,20}] (* _Rigoberto Florez_, May 09 2019 *)
%o (PARI) Vec((1-x)^(-1)/(1+2*x-2*x^2-x^3)+O(x^99)) \\ _Charles R Greathouse IV_, Sep 23 2012
%o (PARI) Vec(1/((1-x)^2*(1+3*x+x^2)) + O(x^100)) \\ _Altug Alkan_, Oct 28 2015
%Y Cf. A002571.
%Y Bisections are A103433 and A103434.
%Y Cf. A064831, A000045. A203803, A207969, A207970.
%K sign,easy
%O 0,3
%A _N. J. A. Sloane_, Nov 17 2002
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