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A202503 Fibonacci self-fission matrix, by antidiagonals. 6
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
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}.
LINKS
Clark Kimberling, Fusion, Fission, and Factors, Fib. Q., 52 (2014), 195-202.
EXAMPLE
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
MATHEMATICA
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 *)
CROSSREFS
Sequence in context: A076272 A180101 A108035 * A049747 A260684 A029093
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Dec 20 2011
STATUS
approved

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Last modified March 28 21:57 EDT 2024. Contains 371254 sequences. (Running on oeis4.)