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

Author

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

Keywords

• sign
• easy

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

• 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).

Crossrefs

Cf. A066170.

Programs

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 *)

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

Extensions

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

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