

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^23x2.
The first entry of the vector v[n]=A*v[n1], 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(n1)+2a(n2) 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[n1]+2*a[n2] 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}] (* or *)
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(n2), 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 and added cross reference.  M. F. Hasler, Dec 01 2008
Edited by N. J. A. Sloane, May 20 2006 and Dec 04 2008


STATUS

approved



