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A077996 Expansion of (1-x)/(1-2*x-x^2-2*x^3). 2

%I

%S 1,1,3,9,23,61,163,433,1151,3061,8139,21641,57543,153005,406835,

%T 1081761,2876367,7648165,20336219,54073337,143779223,382304221,

%U 1016534339,2702931345,7187005471,19110010965,50812890091,135109802089,359252516199,955240614669

%N Expansion of (1-x)/(1-2*x-x^2-2*x^3).

%H G. C. Greubel, <a href="/A077996/b077996.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,2).

%F If p[1]=1, p[2]=2, p[i]=4, (i>2), and if A is Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det A. - _Milan Janjic_, May 02 2010

%F a(n) = Sum_{k=1..n} Sum_{i=k..n} Sum_{j=0..k} binomial(j,-3*k+2*j+i) * 2^(k-j)*binomial(k,j)*binomial(n+k-i-1,k-1). - _Vladimir Kruchinin_, May 05 2011

%t LinearRecurrence[{2,1,2}, {1,1,3}, 40] (* or *) CoefficientList[Series[(1 -x)/(1-2*x-x^2-2*x^3), {x,0,40}], x] (* _G. C. Greubel_, Jun 27 2019 *)

%o (Maxima)

%o a(n):=sum(sum((sum(binomial(j,-3*k+2*j+i)*2^(k-j)*binomial(k,j),j,0,k) )*binomial(n+k-i-1,k-1),i,k,n),k,1,n); /* _Vladimir Kruchinin_, May 05 2011 */

%o (PARI) Vec((1-x)/(1-2*x-x^2-2*x^3)+O(x^40)) \\ _Charles R Greathouse IV_, Sep 27 2012

%o (MAGMA) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x)/(1-2*x-x^2-2*x^3) )); // _G. C. Greubel_, Jun 27 2019

%o (Sage) ((1-x)/(1-2*x-x^2-2*x^3)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Jun 27 2019

%o (GAP) a:=[1,1,3];; for n in [4..30] do a[n]:=2*a[n-1]+a[n-2]+2*a[n-3]; od; a; # _G. C. Greubel_, Jun 27 2019

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_, Nov 17 2002

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Last modified September 20 04:18 EDT 2020. Contains 337264 sequences. (Running on oeis4.)