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Search: id:A109545
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| A109545 |
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5 X 5 vector Markov with Salem characteristic polynomial:-x^5+2*x^4+2*x^3-x^2-x-1. |
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+0
1
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| 1, 1, 2, 6, 15, 38, 97, 247, 629, 1602, 4080, 10391, 26464, 67399, 171653, 437169, 1113390, 2835602, 7221763, 18392518, 46842401, 119299083, 303833085, 773807654, 1970747476, 5019135691, 12782826512, 32555536191, 82913034585
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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a[n]/a[n-1]->2.54682... as n->infinity Alternative vector Markov in Matrix isomer form: a=2;b=-1; M={{0,1,0,0,0},{0,0,1,0,0},{0,0,0,1,0},{0,0,0,0,1},{b,b,-2*a+3,a-2-2*b,a}} v[1]={0,1,1,2,3} v[n_]:=v[n]=M.v[n-1] a0=Table[Abs[v[n][[1]]],{n,1,50}] Det[M - x*IdentityMatrix[5]] b0 = Table[x /. NSolve[Det[M - x*IdentityMatrix[5]] == 0, x][[i]], {i, 1, 5}] Abs[b0] gives a different sequence as the intitial conditions had to be changed. For this sequence the bonacci is: F[1] = 0; F[2] = 1; F[3] = 1; F[4] = 2; F[5] = 3; F[n__] := F[n] = a*F[n - 1] + (a - 2 - 2*b )*F[n - 2] - (2*a - 3)*F[n - 3] + b*F[n - 4] + b*F[n - 5] aa = Table[F[n], {n, 1, 50}]
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REFERENCES
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David Garth and Kevin G. Hare, Comments on the spectra of Pisot numbers. J. Number Theory 121 (2006), 187-203.
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FORMULA
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M = {{0, 1, 0, 0, 0}, {a - 2, a - 2, a - 2 - b, a - 2 - b, 0}, {1, 1, 1, 1, 0}, {0, 1, 1, 0, 0}, {0, 0, 0, 1, 1}} v[n]=M.v[n-1] a(n) = v[n][[1]]
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MATHEMATICA
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a = 2; b = -1; M = {{0, 1, 0, 0, 0}, { a - 2, a - 2, a - 2 - b, a - 2 - b, 0}, {1, 1, 1, 1, 0}, {0, 1, 1, 0, 0}, {0, 0, 0, 1, 1}} v[1] = {1, 1, 1, 1, 1} v[n_] := v[n] = M.v[n - 1] a0 = Table[Abs[v[n][[1]]], {n, 1, 50}]
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CROSSREFS
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Adjacent sequences: A109542 A109543 A109544 this_sequence A109546 A109547 A109548
Sequence in context: A018019 A034518 A106515 this_sequence A120846 A101522 A094969
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KEYWORD
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nonn,uned
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AUTHOR
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Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jun 20 2005
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