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A152459 A posterior vector Markov of A000045 as a triangular sequence: A back iterated Markov with M=Inverse[{{0, 1}, {1, 1}}]={{-1, 1}, {1, 0}}; and v(0)={Fibonacci[n],Fibonacci[n-1]}, to give; t(n,m)=v(m)=(M^m*v(0))_first_element. 0
0, 0, 1, -1, 2, 1, 1, 0, 1, 4, 3, 1, 2, -1, 3, -4, 12, 7, 5, 2, 3, -1, 4, -5, 9, 30, 19, 11, 8, 3, 5, -2, 7, -9, 16, -25, 80, 49, 31, 18, 13, 5, 8, -3, 11, -14, 25, -39, 64, 208, 129, 79, 50, 29, 21, 8, 13, -5, 18, -23, 41, -64, 105, -169, 546, 337, 209, 128, 81, 47, 34, 13, 21, -8, 29 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

The row sums are:

{0, 0, 5, 8, 36, 63, 248, 440, 1701, 3024, 11660,...}.

These row are the result of running a Fibonacci like sequence backward from two

starting points that are Fibonacci. The idea was inspired by the Viterbi posterior

Bayesian type of hidden Markov procedure.

LINKS

Table of n, a(n) for n=0..74.

FORMULA

A back iterated Markov with M=Inverse[{{0, 1}, {1, 1}}]={{-1, 1}, {1, 0}};

and v(0)={Fibonacci[n],Fibonacci[n-1]}, to give;

t(n,m)=v(m)=(M^m*v(0))_first_element.

EXAMPLE

{0},

{0, 1, -1},

{2, 1, 1, 0, 1},

{4, 3, 1, 2, -1, 3, -4},

{12, 7, 5, 2, 3, -1, 4, -5, 9},

{30, 19, 11, 8, 3, 5, -2, 7, -9, 16, -25},

{80, 49, 31, 18, 13, 5, 8, -3, 11, -14, 25, -39, 64},

{208, 129, 79, 50, 29, 21, 8, 13, -5, 18, -23, 41, -64, 105, -169},

{546, 337, 209, 128, 81, 47, 34, 13, 21, -8, 29, -37, 66, -103, 169, -272, 441},

{1428, 883, 545, 338, 207, 131, 76, 55, 21, 34, -13, 47, -60, 107, -167, 274, -441, 715, -1156},

{3740, 2311, 1429, 882, 547, 335, 212, 123, 89, 34, 55, -21, 76, -97, 173, -270, 443, -713, 1156, -1869, 3025}

MATHEMATICA

Clear[M];

M = Inverse[{{0, 1}, {1, 1}}];

Table[Table[(MatrixPower[M, m].{Fibonacci[n], Fibonacci[n - 1]})[[1]], {m, -n, n}], {n, 0, 10}];

Flatten[%]

CROSSREFS

A000045

Sequence in context: A144903 A108934 A108947 * A275784 A097608 A168261

Adjacent sequences:  A152456 A152457 A152458 * A152460 A152461 A152462

KEYWORD

tabf,uned,sign

AUTHOR

Roger L. Bagula, Dec 05 2008

STATUS

approved

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Last modified February 16 21:59 EST 2019. Contains 320200 sequences. (Running on oeis4.)