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A120464
a(n) = 5*a(n-1)+a(n-2)-2*a(n-3).
0
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-x+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).
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
KEYWORD
nonn,easy
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