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A152462
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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)
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0
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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; internal format)
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OFFSET
| 0,4
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COMMENTS
| The row sums are:
{1, -1, 8, 17, 115, 412, 1929, 7771, 33908, 141225, 604359,...}.
This starting vector method gives non-zero low values
and a lower overall triangle.
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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.
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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}
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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[%]
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CROSSREFS
| A000045
Sequence in context: A194543 A027293 A104762 * A180360 A175331 A098805
Adjacent sequences: A152459 A152460 A152461 * A152463 A152464 A152465
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KEYWORD
| tabl,uned,sign
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Dec 05 2008
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