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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).

%I A120758 #14 Apr 15 2025 19:16:54
%S A120758 1,6,25,116,517,2338,10517,47400,213481,961726,4332145,19515036,
%T A120758 87908397,395998298,1783838637,8035595600,36197658961,163058307446,
%U A120758 734522939465,3308779311556,14904940203477,67141752851858,302451060668357,1362440511764600,6137337207120441
%N 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).
%C A120758 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.
%H A120758 Marcel Jackson, <a href="https://web.archive.org/web/20070208121829/http://www.maths.utas.edu.au/People/Jackson/tables/jezek.html">Three Three Element Groupoids of Jezek</a> [archived]
%H A120758 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,7,-1).
%F A120758 a(n) = 3*a(n-1)+7*a(n-2)-a(n-3) (follows from the minimal polynomial of the matrix M).
%F A120758 G.f. x*(1+3*x) / ( 1-3*x-7*x^2+x^3 ). - _R. J. Mathar_, Mar 03 2013
%e A120758 a(7)=10517 because M^7= [6682,9842,10517;9842,14401,15438;10517,15438,16524].
%p A120758 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);
%Y A120758 Cf. A120757.
%K A120758 nonn
%O A120758 1,2
%A A120758 _Gary W. Adamson_ and _Roger L. Bagula_, Jul 01 2006
%E A120758 Corrected by _T. D. Noe_, Nov 07 2006
%E A120758 Edited by _N. J. A. Sloane_, Dec 04 2006