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A202503 Fibonacci self-fission matrix, by antidiagonals.

%I A202503 #20 Jan 05 2025 19:51:39
%S A202503 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 A202503 21,34,37,39,39,39,34,34,34,55,60,63,64,63,63,55,55,55,89,97,102,103,
%U A202503 104,102,102,89,89,89,144,157,165,167,168,168,165,165,144,144,144
%N A202503 Fibonacci self-fission matrix, by antidiagonals.
%C A202503 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 A202503 antidiagonal sums:  (1, 3, 8, 18, 38, ...), A064831
%C A202503 diagonal (1,  5, 14,  39, ...), A119996
%C A202503 diagonal (2,  8, 23,  63, ...), A180664
%C A202503 diagonal (2,  5, 15,  39, ...), A059840
%C A202503 diagonal (3,  8, 24,  63, ...), A080097
%C A202503 diagonal (5, 13, 39, 102, ...), A080143
%C A202503 diagonal (8, 21, 63, 165, ...), A080144
%C A202503 All the principal submatrices are invertible, and the terms in the inverses are in {-3,-2,-1,0,1,2,3}.
%H A202503 Clark Kimberling, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/52-3/Kimberling11132013.pdf">Fusion, Fission, and Factors</a>, Fib. Q., 52 (2014), 195-202.
%e A202503 Northwest corner:
%e A202503 1....1....2....3....5.....8....13...21
%e A202503 2....3....5....8...13....21....34...55
%e A202503 3....5....9...14...23....37....60...97
%e A202503 5....8...15...24...39....63...102...165
%e A202503 8...13...24...39...64...103...167...270
%t A202503 n = 14;
%t A202503 Q = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[Fibonacci[k], {k, 1, n}]];
%t A202503 Qt = Transpose[Q]; P1 = Qt - IdentityMatrix[n];
%t A202503 P = P1[[Range[2, n], Range[1, n]]];
%t A202503 F = P.Q;
%t A202503 Flatten[Table[P[[i]][[k + 1 - i]], {k, 1, n - 1}, {i, 1, k}]] (* A202502 as a sequence *)
%t A202503 Flatten[Table[Q[[i]][[k + 1 - i]], {k, 1, n - 1}, {i, 1, k}]] (* A202451 as a sequence *)
%t A202503 Flatten[Table[F[[i]][[k + 1 - i]], {k, 1, n - 1}, {i, 1, k}]] (* A202503 as a sequence *)
%t A202503 TableForm[P]  (* A202502, modified lower triangular Fibonacci array *)
%t A202503 TableForm[Q]  (* A202451, upper tri. Fibonacci array *)
%t A202503 TableForm[F]  (* A202503, Fibonacci fission array *)
%Y A202503 Cf. A000045, A202451, A202453, A202502.
%K A202503 nonn,tabl
%O A202503 1,3
%A A202503 _Clark Kimberling_, Dec 20 2011