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A114749
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a(n) = a(n-1) + 4a(n-2) + 6a(n-3) + 4a(n-4) + a(n-5).
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1
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0, 1, 1, 2, 3, 21, 50, 161, 501, 1532, 4723, 14551, 44800, 137971, 424901, 1308512, 4029693, 12409831, 38217250, 117693681, 362448951, 1116196192, 3437432913, 10585903361, 32600301650, 100395746291
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| Vector Markov sequence of quartic characteristic Pascal-Salem polynomial x^5-(x+1)^4.
The first three of the sequence of polynomials: x^n-(x+1)^(n-1) are Pisots, this one with two unitary absolute values is Salem r = Abs[Table[x /. NSolve[Det[M - IdentityMatrix[5]*x] == 0, x][[n]], {n, 1, 5}]] gives:{0.56984, 0.56984, 1., 1., 3.0796}
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LINKS
| Index entries for sequences related to linear recurrences with constant coefficients, signature (1,4,6,4,1).
Harvey P. Dale, Table of n, a(n) for n = 0..1000
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FORMULA
| M = {{0, 1, 0, 0, 0}, {0, 0, 1, 0, 0}, {0, 0, 0, 1, 0}, {0, 0, 0, 0, 1}, {1, 4, 6, 4, 1}}; w[0] = {0, 1, 1, 2, 3}; w[n_] := w[n] = M.w[n - 1] a(n) = w[n][[1]]
G.f.:x*(9*x^3+3*x^2-1)/((x^2+x+1)*(x^3+3*x^2+2*x-1)) [From Maksym Voznyy (voznyy(AT)mail.ru), Aug 12 2009]
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MATHEMATICA
| M = {{0, 1, 0, 0, 0}, {0, 0, 1, 0, 0}, {0, 0, 0, 1, 0}, {0, 0, 0, 0, 1}, {1, 4, 6, 4, 1}}; w[0] = {0, 1, 1, 2, 3}; w[n_] := w[n] = M.w[n - 1] a = Flatten[Table[w[n][[1]], {n, 0, 25}]]
LinearRecurrence[{1, 4, 6, 4, 1}, {0, 1, 1, 2, 3}, 30] (* From Harvey P. Dale, Oct 13 2011 *)
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CROSSREFS
| Sequence in context: A124447 A024765 A090122 * A141458 A080670 A073647
Adjacent sequences: A114746 A114747 A114748 * A114750 A114751 A114752
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KEYWORD
| nonn,easy
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AUTHOR
| Roger Bagula (rlbagulatftn(AT)yahoo.com), Feb 18 2006
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