A202503 Fibonacci self-fission matrix, by antidiagonals.

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

Offset: 1

Author

Clark Kimberling, Dec 20 2011

Keywords

• nonn
• tabl

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}.

Examples

			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
		

Links

• Clark Kimberling, Fusion, Fission, and Factors, Fib. Q., 52 (2014), 195-202.

Crossrefs

Cf. A000045, A202451, A202453, A202502.

Programs

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 *)