A120464 a(n) = 5*a(n-1)+a(n-2)-2*a(n-3).

0, 2, 11, 57, 292, 1495, 7653, 39176, 200543, 1026585, 5255116, 26901079, 137707341, 704927552, 3608542943, 18472227585, 94559825764, 484054270519, 2477886723189, 12684368234936, 64931619356831, 332386691572713

Offset

0,2

Comments

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}}.

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

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).

Links

Table of n, a(n) for n=0..21.

Index entries for linear recurrences with constant coefficients, signature (5, 1, -2).

Formula

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]

Mathematica

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}]
LinearRecurrence[{5, 1, -2}, {0, 2, 11}, 30] (* Harvey P. Dale, Sep 25 2017 *)

Crossrefs

Sequence in context: A037490 A037570 A240888 * A164581 A054130 A037738

Adjacent sequences: A120461 A120462 A120463 * A120465 A120466 A120467

Keyword

nonn

Author

Roger L. Bagula, Jul 01 2006

Extensions

Edited by N. J. A. Sloane, Jul 13 2007

Better name by Colin Barker, May 02 2012

Status

approved