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A106434
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The (1,1)-entry of the matrix A^n, where A = [0,1;2,3].
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1
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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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].
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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 (mathar(AT)strw.leidenuniv.nl), 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 (paoloplava(AT)gmail.com), Jun 12 2008
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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);
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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}]
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PROG
| (PARI) A106434(n)=([0, 1; 2, 3]^n)[1, 1] /* M. F. Hasler, Dec 01 2008 */
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CROSSREFS
| Cf. A028860, A100638.
Equals 2*A007482(n-2), for n>1.
Sequence in context: A107239 A148496 A181367 * A150228 A203038 A206304
Adjacent sequences: A106431 A106432 A106433 * A106435 A106436 A106437
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KEYWORD
| nonn
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), May 29 2005
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EXTENSIONS
| Simplified definition, added PARI code and cross reference. - M. F. Hasler (www.univ-ag.fr/~mhasler), Dec 01 2008
Edited by N. J. A. Sloane (njas(AT)research.att.com), May 20 2006 and Dec 04 2008
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