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A123020
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Let M = {{1, 1, 0, 0}, {1, 2, 1, 1}, {0, 1, 2, 1}, {0, 0, 1, 2}}, v[1] = {1, 0, 0, 0}, v[n] = M.v[n - 1]. Then a(n) = v[n][[1]].
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0
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1, 1, 2, 5, 14, 43, 142, 493, 1766, 6443, 23750, 88045, 327406, 1219531, 4546622, 16958765, 63272054, 236096683, 881049142, 3287968813, 12270563966, 45793762763, 170903438510, 637817894125, 2380363943686, 8883629492011
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| A 4 X 4 vector Markov chain with characteristic polynomial 2 - 11 x + 15 x^2 - 7 x^3 + x^4.
Denominator of reduced g.f. is essentially the characteristic polynomial of [1,1,0;1,2,1;0,1,3]. [From Paul Barry (pbarry(AT)wit.ie), Dec 17 2009]
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LINKS
| Eric Weisstein's World of Mathematics, Morgan-Voyce Polynomials
Index to sequences with linear recurrences with constant coefficients, signature (6,-9,2)
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FORMULA
| Contribution from Paul Barry (pbarry(AT)wit.ie), Dec 17 2009: (Start)
G.f.: 1/(1-x-x^2/(1-2x-x^2/(1-3x)))=(1-5x+5x^2)/(1-6x+9x^2-2x^3);
a(n)=(2-sqrt(3))^n*(1/3-sqrt(3)/6)+(2+sqrt(3))^n*(1/3-sqrt(3)/6)+2^n/3. (End)
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MATHEMATICA
| M = {{1, 1, 0, 0}, {1, 2, 1, 1}, {0, 1, 2, 1}, {0, 0, 1, 2}} v[1] = {1, 0, 0, 0} v[n_] := v[n] = M.v[n - 1] a = Table[Floor[v[n][[1]]], {n, 1, 50}]
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CROSSREFS
| Cf. A001519, A080937.
Sequence in context: A066351 A181496 A029889 * A005317 A126566 A112808
Adjacent sequences: A123017 A123018 A123019 * A123021 A123022 A123023
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KEYWORD
| nonn
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AUTHOR
| Roger Bagula and Gary Adamson (rlbagulatftn(AT)yahoo.com), Sep 24 2006
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EXTENSIONS
| Edited by N. J. A. Sloane (njas(AT)research.att.com), Jun 13 2007
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