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A114749 a(n) = a(n-1) + 4*a(n-2) + 6*a(n-3) + 4*a(n-4) + a(n-5). 9
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, 309178300901, 952144142322, 2932218933633 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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}

LINKS

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (1,4,6,4,1).

FORMULA

G.f.: x*(9*x^3 + 3*x^2 - 1)/((x^2 + x + 1)*(x^3 + 3*x^2 + 2*x - 1)). [Maksym Voznyy (voznyy(AT)mail.ru), Aug 12 2009]

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]; Flatten[Table[w[n][[1]], {n, 0, 25}]]

LinearRecurrence[{1, 4, 6, 4, 1}, {0, 1, 1, 2, 3}, 30] (* Harvey P. Dale, Oct 13 2011 *)

PROG

(PARI) x='x+O('x^50); concat([0], Vec(x*(9*x^3+3*x^2-1)/((x^2+x+1)*(x^3+ 3*x^2+2*x-1)))) \\ G. C. Greubel, Nov 03 2018

(MAGMA) m:=50; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(9*x^3+3*x^2-1)/((x^2+x+1)*(x^3+ 3*x^2+2*x-1)))); // G. C. Greubel, Nov 03 2018

CROSSREFS

Cf. A107479, A107480, A109538, A109543, A109544, A125950, A130844, A143335, A147851.

Sequence in context: A090122 A328734 A296254 * A141458 A080670 A288532

Adjacent sequences:  A114746 A114747 A114748 * A114750 A114751 A114752

KEYWORD

nonn,easy

AUTHOR

Roger L. Bagula, Feb 18 2006

STATUS

approved

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Last modified July 8 05:33 EDT 2020. Contains 335513 sequences. (Running on oeis4.)