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A106732 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]. 1
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; text; internal format)
OFFSET

0,2

COMMENTS

Real Pisot roots (the eigenvalues of M): (5-sqrt(13))/2=0.697224,(5+sqrt(13))/2= 4.30278

LINKS

Table of n, a(n) for n=0..22.

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, Oct 07 2008]

a(n) = -3*A116415(n-1), n>0.

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);

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}]

CROSSREFS

Sequence in context: A098102 A144067 A001447 * A052981 A086200 A217451

Adjacent sequences:  A106729 A106730 A106731 * A106733 A106734 A106735

KEYWORD

sign

AUTHOR

Roger L. Bagula, May 30 2005

EXTENSIONS

Edited by N. J. A. Sloane, Apr 30 2006

STATUS

approved

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Last modified September 20 12:42 EDT 2017. Contains 292271 sequences.