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A120775
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The (3,1)-entry of the matrix M^n, where M is the 3 X 3 matrix [0,1,1; 2,1,2; 1,2,2] (n>=1).
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1, 6, 23, 100, 421, 1786, 7563, 32040, 135721, 574926, 2435423, 10316620, 43701901, 185124226, 784198803, 3321919440, 14071876561, 59609425686, 252509579303, 1069647742900, 4531100550901, 19194049946506, 81307300336923
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Characteristic polynomial of M = x^3 - 3x^2 - 5x - 1. a(n)/a(n-1) tends to (2 + sqrt(5)) = phi^3, a root to the characteristic polynomial and an eigenvalue of M.
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LINKS
| Author?, Title
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FORMULA
| a(n) = 3a(n-1 + 5a(n-2) + a(n-3) (follows from the minimal polynomial of the matrix M).
a(n)=(3/4)*[2-sqrt(5)]^n-(1/2)*(-1)^n+(3/4)*[2+sqrt(5)]^n+(1/4)*[2+sqrt(5)]^n*sqrt(5)-(1/4)*[2 -sqrt(5)]^n*sqrt(5), with n>=0 - Paolo P. Lava (paoloplava(AT)gmail.com), Jun 12 2008
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EXAMPLE
| a(6)=1786 because M^6=[799,1045,1292;1596,2091,2584;1786,2337,2889].
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MAPLE
| with(linalg): M[1]:=matrix(3, 3, [0, 1, 1, 2, 1, 2, 1, 2, 2]): for n from 2 to 25 do M[n]:=multiply(M[1], M[n-1]) od: seq(M[n][3, 1], n=1..25);
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CROSSREFS
| Cf. A120758, A120757.
Sequence in context: A013261 A013265 A038383 * A013258 A013264 A063383
Adjacent sequences: A120772 A120773 A120774 * A120776 A120777 A120778
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KEYWORD
| nonn
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AUTHOR
| Gary W. Adamson and Roger L. Bagula (qntmpkt(AT)yahoo.com), Jul 04 2006
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EXTENSIONS
| Edited by N. J. A. Sloane (njas(AT)research.att.com), Dec 03 2006
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