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A106732
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First entry of the vector (M^n)v, where M is the 2 X 2 matrix [[0,-3],[1,5]] and v is the column vector [0,1].
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0
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0, -3, -15, -66, -285, -1227, -5280, -22719, -97755, -420618, -1809825, -7787271, -33506880, -144172587, -620342295, -2669193714, -11484941685, -49417127283, -212630811360, -914902674951, -3936620940675, -16938396678522, -72882120570585
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Real Pisot roots (the eigenvalues of M): (5-sqrt(13))/2=0.697224,(5+sqrt(13))/2= 4.30278
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FORMULA
| a(n)=first entry of v[n], where v[n]=Mv[n-1], M is the 2 X 2 matrix [[0, -3], [1, 5]] and v[0] is the column vector [0,1]. G.f.=-3x/(1-5x+3x^2). a(n)=5a(n-1)-3a(n-2); a(0)=0, a(1)=-3.
a(n)=(3/13)*[5/2-(1/2)*sqrt(13)]^n*sqrt(13)-(3/13)*sqrt(13)*[5/2+(1/2)*sqrt(13)]^n, with n>=0 [From Paolo P. Lava (paoloplava(AT)gmail.com), Oct 07 2008]
a(n) = -3*A116415(n-1), n>0.
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MAPLE
| a[0]:=0: a[1]:=-3: for n from 2 to 22 do a[n]:=5*a[n-1]-3*a[n-2] od: seq(a[n], n=0..22);
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MATHEMATICA
| M = {{0, -3}, {1, 5}} 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
| Sequence in context: A098102 A144067 A001447 * A052981 A086200 A122558
Adjacent sequences: A106729 A106730 A106731 * A106733 A106734 A106735
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KEYWORD
| sign
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), May 30 2005
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EXTENSIONS
| Edited by N. J. A. Sloane (njas(AT)research.att.com), Apr 30 2006
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