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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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5
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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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a(n) is the 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].
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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G. C. Greubel, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (4,-1).
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FORMULA
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G.f.: -x/(1-4*x+x^2).
a(n) = 4*a(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. - 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}]
Table[Round[(1/6)*Sqrt[3]*(2 - Sqrt[3])^n - (1/6)*Sqrt[3]*(2 + Sqrt[3])^n ], {n, 0, 50}] (* G. C. Greubel, Feb 05 2018 *)
LinearRecurrence[{4, -1}, {0, -1}, 30] (* Harvey P. Dale, Nov 01 2019 *)
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PROG
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(PARI) x='x+O('x^30); Vec(-x/(1-4*x+x^2)) \\ G. C. Greubel, Feb 05 2018
(Magma) I:=[0, -1]; [n le 2 select I[n] else 4*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Feb 05 2018
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CROSSREFS
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Cf. A001076, A001353.
Sequence in context: A242495 A221859 A001353 * A125905 A195503 A010905
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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