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A106434 The (1,1)-entry of the matrix A^n, where A = [0,1;2,3]. 2
0, 2, 6, 22, 78, 278, 990, 3526, 12558, 44726, 159294, 567334, 2020590, 7196438, 25630494, 91284358, 325114062, 1157910902, 4123960830, 14687704294, 52311034542, 186308512214, 663547605726, 2363259841606, 8416874736270, 29977143892022, 106765181148606 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The characteristic polynomial of the matrix A is x^2-3x-2.

The first entry of the vector v[n]=Av[n-1], where A is the 2 X 2 matrix [[0,2],[1,3]] and v[1] is the column vector [0,1].

The (1,1)-entry of the matrix A^n where A=[0,1,1;1,2,1;1,1,2]. - David Neil McGrath, Jul 18 2014

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (3,2).

FORMULA

Recurrence relation: a(n)=3a(n-1)+2a(n-2) for n>=3; a(1)=0, a(2)=2.

O.g.f.: -2*x^2/(-1+3*x+2*x^2). - R. J. Mathar, Dec 05 2007

a(n)=-(2/17)*sqrt(17)*[3/2-(1/2)*sqrt(17)]^n+(2/17)*[3/2+(1/2)*sqrt(17)]^n*sqrt(17), with n>=0 - Paolo P. Lava, Jun 12 2008

MAPLE

a[1]:=0: a[2]:=2: for n from 3 to 25 do a[n]:=3*a[n-1]+2*a[n-2] od: seq(a[n], n=1..25);

MATHEMATICA

M = {{0, 2}, {1, 3}} v[1] = {0, 1} v[n_] := v[n] = M.v[n - 1] a = Table[Abs[v[n][[1]]], {n, 1, 50}] (* Roger L. Bagula *)

LinearRecurrence[{3, 2}, {0, 2}, 50] (* Vladimir Joseph Stephan Orlovsky, Feb 24 2012 *)

PROG

(PARI) A106434(n)=([0, 1; 2, 3]^n)[1, 1] /* M. F. Hasler, Dec 01 2008 */

CROSSREFS

Cf. A028860, A100638.

Equals 2*A007482(n-2), for n>1.

Sequence in context: A148496 A217528 A181367 * A150228 A203038 A206304

Adjacent sequences:  A106431 A106432 A106433 * A106435 A106436 A106437

KEYWORD

nonn,easy

AUTHOR

Roger L. Bagula, May 29 2005

EXTENSIONS

Simplified definition, added PARI code and cross reference. - M. F. Hasler, Dec 01 2008

Edited by N. J. A. Sloane, May 20 2006 and Dec 04 2008

STATUS

approved

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Last modified November 23 21:46 EST 2017. Contains 295141 sequences.