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A122187

First row sum of the matrix M^n, where M is the 3 X 3 matrix [[6, 5, 3], [5, 4, 2], [3, 2, 1]] (n>=0).

1, 14, 157, 1782, 20216, 229347, 2601899, 29518061, 334876920, 3799116465, 43100270734, 488964567014, 5547212203625, 62932092237197, 713952898856653, 8099663044168346, 91889172989041221, 1042465602157270162

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OFFSET

1,2

LINKS

Table of n, a(n) for n=1..18.

P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.

Index entries for linear recurrences with constant coefficients, signature (11,4,-1).

FORMULA

a(n) = 11a(n-1)+4a(n-2)-a(n-3), a(0)=1, a(1)=14, a(2)=157 (derived from the minimal polynomial of the matrix M).

G.f.: -x*(x^2-3*x-1) / (x^3-4*x^2-11*x+1). [Colin Barker, Dec 07 2012]

MAPLE

a[0]:=1: a[1]:=14: a[2]:=157: for n from 3 to 20 do a[n]:=11*a[n-1]+4*a[n-2]-a[n-3] od: seq(a[n], n=0..20);

MATHEMATICA

M = {{6, 5, 3}, {5, 4, 2}, {3, 2, 1}}; v[1] = {1, 1, 1}; v[n_] := v[n] = M.v[n - 1]; a1 = Table[v[n][[1]], {n, 1, 50}]

LinearRecurrence[{11, 4, -1}, {1, 14, 157}, 30] (* Harvey P. Dale, Oct 18 2022 *)

CROSSREFS

Cf. A015448.

Sequence in context: A238770 A199703 A144166 * A268946 A383946 A343093

Adjacent sequences: A122184 A122185 A122186 * A122188 A122189 A122190

KEYWORD

nonn,easy

AUTHOR

Gary W. Adamson and Roger L. Bagula, Oct 17 2006

EXTENSIONS

Edited by N. J. A. Sloane, Oct 29 2006

STATUS

approved

URL: https://oeis.org/A122187

⇱ A122187 - OEIS