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 A152462 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. (Starting vector symmetrical in n,m.) 0
 1, 1, -1, 2, -1, 1, 3, -2, 1, -1, 5, -3, 3, -1, -1, 8, -5, 5, -4, -1, 7, 13, -8, 9, -7, 4, 7, -27, 21, -13, 15, -13, 9, -1, -27, 83, 34, -21, 25, -22, 19, -9, -14, 83, -239, 55, -34, 41, -37, 34, -25, -1, 62, -239, 659, 89, -55, 67, -61, 59, -49, 25, 41, -205, 659, -1781 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS The row sums are {1, -1, 8, 17, 115, 412, 1929, 7771, 33908, 141225, 604359, ...}. This starting vector method gives nonzero low values and a lower overall triangle. 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 {1}, {1, -1}, {2, -1, 1}, {3, -2, 1, -1}, {5, -3, 3, -1, -1}, {8, -5, 5, -4, -1, 7}, {13, -8, 9, -7, 4, 7, -27}, {21, -13, 15, -13, 9, -1, -27, 83}, {34, -21, 25, -22, 19, -9, -14,83, -239}, {55, -34, 41, -37, 34, -25, -1, 62, -239, 659}, {89, -55, 67, -61, 59, -49, 25, 41, -205, 659, -1781} MATHEMATICA Clear[M, a]; M = Inverse[{{0, 1}, {1, 1}}]; a = Table[(MatrixPower[M, n].{1, 0})[[1]], {n, -30, 30}]; Table[Table[(MatrixPower[M, m].{a[[30 - (n - m + 1)]], a[[30 - (m - 1)]]})[[1]], {m, 0, n}], {n, 0, 10}]; Flatten[%] CROSSREFS Cf. A000045. Sequence in context: A287920 A027293 A104762 * A180360 A318805 A175331 Adjacent sequences:  A152459 A152460 A152461 * A152463 A152464 A152465 KEYWORD tabl,uned,sign AUTHOR Roger L. Bagula, Dec 05 2008 STATUS approved

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Last modified May 16 14:39 EDT 2021. Contains 343949 sequences. (Running on oeis4.)