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A202503 Fibonacci self-fission matrix, by antidiagonals. 6

%I

%S 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,

%T 21,34,37,39,39,39,34,34,34,55,60,63,64,63,63,55,55,55,89,97,102,103,

%U 104,102,102,89,89,89,144,157,165,167,168,168,165,165,144,144,144

%N Fibonacci self-fission matrix, by antidiagonals.

%C 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.

%C antidiagonal sums: (1, 3, 8, 18, 38, ...), A064831

%C diagonal (1, 5, 14, 39, ...), A119996

%C diagonal (2, 8, 23, 63, ...), A180664

%C diagonal (2, 5, 15, 39, ...), A059840

%C diagonal (3, 8, 24, 63, ...), A080097

%C diagonal (5, 13, 39, 102, ...), A080143

%C diagonal (8, 21, 63, 165, ...), A080144

%C All the principal submatrices are invertible, and the terms in the inverses are in {-3,-2,-1,0,1,2,3}.

%H Clark Kimberling, <a href="https://www.fq.math.ca/Papers1/52-3/Kimberling11132013.pdf">Fusion, Fission, and Factors</a>, Fib. Q., 52 (2014), 195-202.

%e Northwest corner:

%e 1....1....2....3....5.....8....13...21

%e 2....3....5....8...13....21....34...55

%e 3....5....9...14...23....37....60...97

%e 5....8...15...24...39....63...102...165

%e 8...13...24...39...64...103...167...270

%t n = 14;

%t Q = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[Fibonacci[k], {k, 1, n}]];

%t Qt = Transpose[Q]; P1 = Qt - IdentityMatrix[n];

%t P = P1[[Range[2, n], Range[1, n]]];

%t F = P.Q;

%t Flatten[Table[P[[i]][[k + 1 - i]], {k, 1, n - 1}, {i, 1, k}]] (* A202502 as a sequence *)

%t Flatten[Table[Q[[i]][[k + 1 - i]], {k, 1, n - 1}, {i, 1, k}]] (* A202451 as a sequence *)

%t Flatten[Table[F[[i]][[k + 1 - i]], {k, 1, n - 1}, {i, 1, k}]] (* A202503 as a sequence *)

%t TableForm[P] (* A202502, modified lower triangular Fibonacci array *)

%t TableForm[Q] (* A202451, upper tri. Fibonacci array *)

%t TableForm[F] (* A202503, Fibonacci fission array *)

%Y Cf. A000045, A202451, A202453, A202502.

%K nonn,tabl

%O 1,3

%A _Clark Kimberling_, Dec 20 2011

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Last modified April 5 22:25 EDT 2020. Contains 333260 sequences. (Running on oeis4.)