

A120464


a(n) = 5*a(n1)+a(n2)2*a(n3).


0



0, 2, 11, 57, 292, 1495, 7653, 39176, 200543, 1026585, 5255116, 26901079, 137707341, 704927552, 3608542943, 18472227585, 94559825764, 484054270519, 2477886723189, 12684368234936, 64931619356831, 332386691572713
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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^35*x^2x+2. Roots: {0.6874, 0.568373, 5.11903}. Ratio: 5.11903.
Lyndon (1951) earlier had proved every twoelement algebra has a finitely based system of identities. However Murskii (1965) found this classic 3element example (which is inherently not finitely based).


LINKS



FORMULA

a(n) = 5*a(n1)+a(n2)2*a(n3). G.f.: x*(2+x)/(15*xx^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



KEYWORD

nonn,easy,changed


AUTHOR



EXTENSIONS



STATUS

approved



