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A077996
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Expansion of (1-x)/(1-2*x-x^2-2*x^3).
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4
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1, 1, 3, 9, 23, 61, 163, 433, 1151, 3061, 8139, 21641, 57543, 153005, 406835, 1081761, 2876367, 7648165, 20336219, 54073337, 143779223, 382304221, 1016534339, 2702931345, 7187005471, 19110010965, 50812890091, 135109802089, 359252516199, 955240614669
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OFFSET
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0,3
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COMMENTS
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a(n) is also the number of ways to tile a long box of dimensions 2 X 2 X n with "plates" (of dimension 2 X 2 X 1) and "hinges" (which are 2 X 2 X 1 plates with a 2 X 1 X 1 box added on top or bottom of the plate). The plates and hinges are shown here:
______ ______
/ /| and / /|
/_____/ / /_____/ |
|_____|/ |___ | /
|_|/
(End)
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LINKS
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FORMULA
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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
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
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MATHEMATICA
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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 *)
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PROG
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(Maxima)
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 */
(Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x)/(1-2*x-x^2-2*x^3) )); // G. C. Greubel, Jun 27 2019
(Sage) ((1-x)/(1-2*x-x^2-2*x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jun 27 2019
(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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