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A106707
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First entry of the vector (M^n)v, where M is the 2 X 2 matrix [[0,-1],[1,4]] and v is the column vector [0,1].
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3
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0, -1, -4, -15, -56, -209, -780, -2911, -10864, -40545, -151316, -564719, -2107560, -7865521, -29354524, -109552575, -408855776, -1525870529, -5694626340, -21252634831, -79315912984, -296011017105, -1104728155436, -4122901604639, -15386878263120, -57424611447841
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OFFSET
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0,3
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COMMENTS
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Real Pisot roots (the eigenvalues of M): 2-sqrt(3)=0.267949, 2+sqrt(3)=3.73205.
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LINKS
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Table of n, a(n) for n=0..25.
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (4,-1).
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FORMULA
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a(n)=first entry of v[n], where v[n]=Mv[n-1], M is the 2 X 2 matrix [[0, -1], [1, 4]] and v[0] is the column vector [0,1]. G.f.=-x/(1-4x+x^2). a(n)=4a(n-1)-a(n-2); a(0)=0, a(1)=-1.
a(n)=(1/6)*sqrt(3)*[2-sqrt(3)]^n-(1/6)*sqrt(3)*[2+sqrt(3)]^n, with n>=0 [From Paolo P. Lava, Oct 06 2008]
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MAPLE
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a[0]:=0: a[1]:=-1: for n from 2 to 27 do a[n]:=4*a[n-1]-a[n-2] od: seq(a[n], n=0..27);
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MATHEMATICA
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M = {{0, -1}, {1, 4}} v[1] = {0, 1} v[n_] := v[n] = M.v[n - 1] a = Table[Abs[v[n][[1]]], {n, 1, 50}]
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CROSSREFS
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Cf. A001076, A001353.
Sequence in context: A001353 * A125905 A195503 A010905 A026030 A047038
Adjacent sequences: A106704 A106705 A106706 * A106708 A106709 A106710
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KEYWORD
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sign,easy
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AUTHOR
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Roger L. Bagula, May 30 2005
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EXTENSIONS
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Edited by N. J. A. Sloane, Apr 30 2006
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STATUS
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approved
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