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A202503
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Fibonacci self-fission matrix, by antidiagonals.
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6
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1, 1, 2, 2, 3, 3, 3, 5, 5, 5, 5, 8, 9, 8, 8, 8, 13, 14, 15, 13, 13, 13, 21, 23, 24, 24, 21, 21, 21, 34, 37, 39, 39, 39, 34, 34, 34, 55, 60, 63, 64, 63, 63, 55, 55, 55, 89, 97, 102, 103, 104, 102, 102, 89, 89, 89, 144, 157, 165, 167, 168, 168, 165, 165, 144, 144, 144
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OFFSET
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1,3
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COMMENTS
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The Fibonacci self-fission matrix, F, is the fission P^^Q, where P and Q are the matrices given at A202502 and A202451. See A193842 for the definition of fission.
antidiagonal sums: (1, 3, 8, 18, 38, ...), A064831
diagonal (1, 5, 14, 39, ...), A119996
diagonal (2, 8, 23, 63, ...), A180664
diagonal (2, 5, 15, 39, ...), A059840
diagonal (3, 8, 24, 63, ...), A080097
diagonal (5, 13, 39, 102, ...), A080143
diagonal (8, 21, 63, 165, ...), A080144
All the principal submatrices are invertible, and the terms in the inverses are in {-3,-2,-1,0,1,2,3}.
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LINKS
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EXAMPLE
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Northwest corner:
1....1....2....3....5.....8....13...21
2....3....5....8...13....21....34...55
3....5....9...14...23....37....60...97
5....8...15...24...39....63...102...165
8...13...24...39...64...103...167...270
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MATHEMATICA
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n = 14;
Q = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[Fibonacci[k], {k, 1, n}]];
Qt = Transpose[Q]; P1 = Qt - IdentityMatrix[n];
P = P1[[Range[2, n], Range[1, n]]];
F = P.Q;
Flatten[Table[P[[i]][[k + 1 - i]], {k, 1, n - 1}, {i, 1, k}]] (* A202502 as a sequence *)
Flatten[Table[Q[[i]][[k + 1 - i]], {k, 1, n - 1}, {i, 1, k}]] (* A202451 as a sequence *)
Flatten[Table[F[[i]][[k + 1 - i]], {k, 1, n - 1}, {i, 1, k}]] (* A202503 as a sequence *)
TableForm[P] (* A202502, modified lower triangular Fibonacci array *)
TableForm[Q] (* A202451, upper tri. Fibonacci array *)
TableForm[F] (* A202503, Fibonacci fission array *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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