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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: A108934 A108947 A338859 * A275784 A331508 A097608

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 May 14 06:28 EDT 2021. Contains 343879 sequences. (Running on oeis4.)