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A106434 The (1,1)-entry of the matrix A^n, where A = [0,1;2,3]. 2

%I

%S 0,2,6,22,78,278,990,3526,12558,44726,159294,567334,2020590,7196438,

%T 25630494,91284358,325114062,1157910902,4123960830,14687704294,

%U 52311034542,186308512214,663547605726,2363259841606,8416874736270,29977143892022,106765181148606

%N The (1,1)-entry of the matrix A^n, where A = [0,1;2,3].

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

%C 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].

%C 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

%H Vincenzo Librandi, <a href="/A106434/b106434.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,2).

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

%F O.g.f.: -2*x^2/(-1+3*x+2*x^2). - _R. J. Mathar_, Dec 05 2007

%F 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

%p 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);

%t 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_ *)

%t LinearRecurrence[{3, 2}, {0, 2}, 50] (* _Vladimir Joseph Stephan Orlovsky_, Feb 24 2012 *)

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

%Y Cf. A028860, A100638.

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

%K nonn,easy

%O 1,2

%A _Roger L. Bagula_, May 29 2005

%E Simplified definition, added PARI code and cross reference. - _M. F. Hasler_, Dec 01 2008

%E Edited by _N. J. A. Sloane_, May 20 2006 and Dec 04 2008

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Last modified March 19 17:21 EDT 2019. Contains 321330 sequences. (Running on oeis4.)