A120758 The (1,3)-entry in the matrix M^n, where M is the 3 X 3 matrix [0,2,1; 2,1,2; 1,2,2] (n>=1).

1, 6, 25, 116, 517, 2338, 10517, 47400, 213481, 961726, 4332145, 19515036, 87908397, 395998298, 1783838637, 8035595600, 36197658961, 163058307446, 734522939465, 3308779311556, 14904940203477, 67141752851858, 302451060668357, 1362440511764600, 6137337207120441

Offset

1,2

Comments

a(n)/a(n-1) tends to 4.50466435...an eigenvalue of M and a root to the characteristic polynomial x^3 - 3x^2 - 7x + 1.

Links

Table of n, a(n) for n=1..25.

Marcel Jackson, Three Three Element Groupoids of Jezek [archived]

Index entries for linear recurrences with constant coefficients, signature (3,7,-1).

Formula

a(n) = 3*a(n-1)+7*a(n-2)-a(n-3) (follows from the minimal polynomial of the matrix M).

G.f. x*(1+3*x) / ( 1-3*x-7*x^2+x^3 ). - R. J. Mathar, Mar 03 2013

Example

a(7)=10517 because M^7= [6682,9842,10517;9842,14401,15438;10517,15438,16524].

Maple

with(linalg): M[1]:=matrix(3, 3, [0, 2, 1, 2, 1, 2, 1, 2, 2]): for n from 2 to 25 do M[n]:=multiply(M[1], M[n-1]) od: seq(M[n][3, 1], n=1..25);

Prog

(PARI) a(n)=([0, 1, 0; 0, 0, 1; -1, 7, 3]^(n-1)*[1; 6; 25])[1, 1] \\ Charles R Greathouse IV, Jun 03 2026

Crossrefs

Cf. A120757.

Sequence in context: A094669 A100296 A346818 * A227914 A179603 A298700

Adjacent sequences: A120755 A120756 A120757 * A120759 A120760 A120761

Keyword

nonn,easy,changed

Author

Gary W. Adamson and Roger L. Bagula, Jul 01 2006

Extensions

Corrected by T. D. Noe, Nov 07 2006

Edited by N. J. A. Sloane, Dec 04 2006

Status

approved