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A202453 Fibonacci self-fusion matrix, by antidiagonals.

%I A202453 #24 Jan 05 2025 19:51:39
%S A202453 1,1,1,2,2,2,3,3,3,3,5,5,6,5,5,8,8,9,9,8,8,13,13,15,15,15,13,13,21,21,
%T A202453 24,24,24,24,21,21,34,34,39,39,40,39,39,34,34,55,55,63,63,64,64,63,63,
%U A202453 55,55,89,89,102,102,104,104,104,102,102,89,89,144,144,165,165
%N A202453 Fibonacci self-fusion matrix, by antidiagonals.
%C A202453 The Fibonacci self-fusion matrix, F, is the fusion P**Q, where P and Q are the lower and upper triangular Fibonacci matrices.  See A193722 for the definition of fusion of triangular arrays.
%C A202453 Every term F(n,k) of F is a product of two Fibonacci numbers; indeed,
%C A202453 F(n,k)=F(n)*F(k+1) if k is even;
%C A202453 F(n,k)=F(n+1)*F(k) if k is odd.
%C A202453 antidiagonal sums: (1,2,6,12,...), A054454
%C A202453 diagonal (1,2,6,15,...), A001654
%C A202453 diagonal (1,3,9,24,...), A064831
%C A202453 diagonal (2,5,15,39,..), A059840
%C A202453 diagonal (3,8,24,63,..), A080097
%C A202453 diagonal (5,13,39,102,...), A080143
%C A202453 diagonal (8,21,63,165,...), A080144
%C A202453 principal submatrix sums, A202462
%C A202453 All the principal submatrices are invertible, and the terms in the inverses are in {-3,-2,-1,0,1,2,3}.
%H A202453 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.
%F A202453 Matrix product P*Q, where P, Q are the lower and upper triangular Fibonacci matrices, A202451 and A202452.
%e A202453 Northwest corner:
%e A202453 1...1....2....3....5....8....13
%e A202453 1...2....3....5....8...13....21
%e A202453 2...3....6....9...15...24....39
%e A202453 3...5....9...15...24...39....63
%e A202453 5...8...15...24...40...64...104
%t A202453 n = 12;
%t A202453 Q = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[Fibonacci[k], {k, 1, n}]];
%t A202453 P = Transpose[Q]; F = P.Q;
%t A202453 Flatten[Table[P[[i]][[k + 1 - i]], {k, 1, n}, {i, 1, k}]] (* A202452 as a sequence *)
%t A202453 Flatten[Table[Q[[i]][[k + 1 - i]], {k, 1, n}, {i, 1, k}]] (* A202451 as a sequence *)
%t A202453 Flatten[Table[F[[i]][[k + 1 - i]], {k, 1, n}, {i, 1, k}]] (* A202453 as a sequence *)
%t A202453 TableForm[Q]  (* A202451, upper tri. Fibonacci array *)
%t A202453 TableForm[P]  (* A202452, lower tri. Fibonacci array *)
%t A202453 TableForm[F]  (* A202453, Fibonacci fusion array *)
%t A202453 TableForm[FactorInteger[F]]
%Y A202453 Cf. A000045, A202451, A202452, A202503 (Fibonacci fission array).
%K A202453 nonn,tabl
%O A202453 1,4
%A A202453 _Clark Kimberling_, Dec 19 2011