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A106511 Expansion of (1+x)^2/((1 + x + x^2)*(1 + x - x^2)). 3

%I

%S 1,0,0,0,1,-2,3,-4,6,-10,17,-28,45,-72,116,-188,305,-494,799,-1292,

%T 2090,-3382,5473,-8856,14329,-23184,37512,-60696,98209,-158906,257115,

%U -416020,673134,-1089154,1762289,-2851444,4613733,-7465176,12078908,-19544084,31622993

%N Expansion of (1+x)^2/((1 + x + x^2)*(1 + x - x^2)).

%C Diagonal sums of the Riordan array ((1+x)/(1+x+x^2), x/(1+x)), A106509.

%H Vincenzo Librandi, <a href="/A106511/b106511.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 (-2,-1,0,1).

%F a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-2k} (-1)^j*binomial(2n-3k-j, j).

%F a(n) = (1/2)*((-1)^n*Fibonacci(n) + Kronecker(-3,n)). - _Ralf Stephan_, Jun 02 2007

%F a(n) = -2*a(n-1) - a(n-2) + a(n-4), a(0)=1, a(1)=a(2)=a(3)=0. - _Philippe Deléham_, Jan 12 2014

%F a(n) = (-1)^n*(Fibonacci(n-1) - A005252(n-2)), n>=2. - _Katharine Ahrens_, May 05 2019

%t CoefficientList[Series[(1 + x)^2/((1 + x + x^2)(1 + x - x^2)), {x, 0, 40}], x] (* _Vincenzo Librandi_, Jan 12 2014 *)

%o (MAGMA) I:=[1,0,0,0]; [n le 4 select I[n] else -2*Self(n-1)-Self(n-2)+Self(n-4): n in [1..50]]; // _Vincenzo Librandi_, Jan 12 2014

%Y Cf. A024490, A011646, A039834.

%K easy,sign

%O 0,6

%A _Paul Barry_, May 04 2005

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Last modified September 20 05:51 EDT 2019. Contains 327212 sequences. (Running on oeis4.)