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).
LINKS
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
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