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A130619
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A doubly recursive vector matrix Markov with charactersistic polynomial: 1 + n x^2 - x^3 Based on matrices of the form: M(n)={{n, 0, 1}, {1, 0, 0}, {0, 1, 0}};.
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0
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0, 1, 2, 6, 25, 127, 768, 5401, 43335, 390783, 3913231, 43088876, 517457295, 6730858066, 94275101800, 1414643984295, 22641034606786, 384991863417162, 6931268185493211, 131716736558977795, 2634719723042973062
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| I call this the minimal Pisot based transvective sequence. Transvection is a matrix type in group theory: It is an identity matrix plus one off diagonal term: T(i,j,a)=Ident+E(i,j,a) These matrices have a couple of unique properties:1) 1) inverse T(i,j,a)^(-1)=Ident+E(i,j,-a) 2) matrix power T(i,j,a)^(m)=Ident+E(i,j,m*a) If you look at the matrix for the minimal Pisot polynomial: ( which I have called a pseudo-permutaion matrix before) M = {{0, 1, 0}, {0, 0, 1}, {1, 1, 0}} CharacteristicPolynomial[M, x] 1 + x - x^3 MatrixPower[M, -1] {{-1, 0, 1}, {1, 0, 0}, {0, 1, 0}} So the matrix power equivalent would just be the matrix M(n) I am using. It involves a new way to look at the algebra involved in a Markov sequence.
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FORMULA
| M(0)={{0, 1, 0}, {0, 0, 1}, {1, 1, 0}}; M(n)={{n, 0, 1}, {1, 0, 0}, {0, 1, 0}}; v(n)=M(n)*v(n-1) a(n) = v(n-1)[[1]]
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MATHEMATICA
| M[0] = {{0, 1, 0}, {0, 0, 1}, {1, 1, 0}}; M[n_] := {{n, 0, 1}, {1, 0, 0}, {0, 1, 0}}; v[0] = {0, 0, 1}; v[n_] := v[n] = M[n].v[n - 1]; a = Table[v[n][[1]], {n, 0, 30}]
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CROSSREFS
| Sequence in context: A030883 A030899 A030907 * A181594 A030915 A030921
Adjacent sequences: A130616 A130617 A130618 * A130620 A130621 A130622
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KEYWORD
| nonn,uned
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jun 18 2007
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