A120464 - OEIS

%I #27 Sep 14 2024 06:51:40

%S 0,2,11,57,292,1495,7653,39176,200543,1026585,5255116,26901079,

%T 137707341,704927552,3608542943,18472227585,94559825764,484054270519,

%U 2477886723189,12684368234936,64931619356831,332386691572713

%N a(n) = 5*a(n-1)+a(n-2)-2*a(n-3).

%C Old name was: Sequence produced by 3 X 3 Markov chain based on Murskii's Cayley table for a three element groupoid: M = {{1,1,1},{1,1,1},{1,1,1}}+{{0,0,0},{0,0,1},{0,2,2}} = {{1, 1, 1}, {1, 1, 2}, {1, 3, 3}}.

%C Characteristic polynomial x^3-5*x^2-x+2. Roots: {-0.6874, 0.568373, 5.11903}. Ratio: 5.11903.

%C Lyndon (1951) earlier had proved every two-element algebra has a finitely based system of identities. However Murskii (1965) found this classic 3-element example (which is inherently not finitely based).

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5, 1, -2).

%F a(n) = 5*a(n-1)+a(n-2)-2*a(n-3). G.f.: x*(2+x)/(1-5*x-x^2+2*x^3). - _Colin Barker_, May 02 2012

%t M = {{1, 1, 1}, {1, 1, 2}, {1, 3, 3}} v[1] = {0, 1, 1} v[n_] := v[n] = M.v[n - 1] a = Table[Floor[v[n][[1]]], {n, 1, 50}] Det[M - x*IdentityMatrix[3]] Factor[%] aaa = Table[x /. NSolve[Det[M - x*IdentityMatrix[3]] == 0, x][[n]], {n, 1, 3}] Abs[aaa] a1 = Table[N[a[[n]]/a[[n - 1]]], {n, 7, 50}]

%t LinearRecurrence[{5,1,-2},{0,2,11},30] (* _Harvey P. Dale_, Sep 25 2017 *)

%K nonn,easy

%O 0,2

%A _Roger L. Bagula_, Jul 01 2006

%E Edited by _N. J. A. Sloane_, Jul 13 2007

%E Better name by _Colin Barker_, May 02 2012