A122607 Expansion of x*(8*x^5 + 5*x^4 - x^3 - 5*x^2 - 1)/(x^6 + 3*x^5 + 6*x^4 + 4*x^3 - 5*x^2 + x - 1).

1, 1, 1, 1, 1, 1, 10, 19, -17, -62, 163, 550, -548, -3050, 2665, 19450, -7550, -113534, 8308, 667423, 187462, -3800825, -2366747, 21303154, 21068938, -116488961, -162036530, 621601885, 1153785034, -3216794309, -7799929064, 16026195376, 50784142789, -75764359214, -320876463932

Offset

1,7

Comments

Obtained as the top element of the vector resulting from multiplying the n-th power of the 6 X 6 matrix [[0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1], [1, 3, 6, 4, -5, 1]] with the column vector which contains only 1's.

Links

Table of n, a(n) for n=1..35.

Peter Steinbach, Golden fields: a case for the heptagon, Math. Mag. Vol. 70, No. 1, Feb. 1997, 22-31.

Index entries for linear recurrences with constant coefficients, signature (1,-5,4,6,3,1).

Formula

G.f.: x*(8*x^5+5*x^4-x^3-5*x^2-1)/(x^6+3*x^5+6*x^4+4*x^3-5*x^2+x-1). - Colin Barker, Nov 08 2012

Mathematica

M = {{0, 1, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0}, {0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 1}, {1, 3, 6, 4, -5, 1}}; v[1] = {1, 1, 1, 1, 1, 1}; v[n_] := v[n] = M.v[n - 1] a = Table[Floor[v[n][[1]]], {n, 1, 50}]
LinearRecurrence[{1, -5, 4, 6, 3, 1}, {1, 1, 1, 1, 1, 1}, 40] (* Harvey P. Dale, Feb 17 2024 *)

Crossrefs

Cf. A066170.

Sequence in context: A240022 A264386 A173822 * A113703 A103757 A287058

Adjacent sequences: A122604 A122605 A122606 * A122608 A122609 A122610

Keyword

sign,easy

Author

Roger L. Bagula and Gary W. Adamson, Sep 20 2006

Extensions

Edited by N. J. A. Sloane, Sep 24 2006

Definition changed using Barker's g.f. by Bruno Berselli, Sep 19 2017

Status

approved