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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 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 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.)