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

1,3

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

Table of n, a(n) for n=1..67.

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

Cf. A000045, A202451, A202453, A202502.

Sequence in context: A076272 A180101 A108035 * A049747 A260684 A029093

Adjacent sequences: A202500 A202501 A202502 * A202504 A202505 A202506

Keyword

nonn,tabl

Author

Clark Kimberling, Dec 20 2011

Status

approved