A202453 Fibonacci self-fusion matrix, by antidiagonals.

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, 24, 24, 24, 24, 21, 21, 34, 34, 39, 39, 40, 39, 39, 34, 34, 55, 55, 63, 63, 64, 64, 63, 63, 55, 55, 89, 89, 102, 102, 104, 104, 104, 102, 102, 89, 89, 144, 144, 165, 165

Offset

1,4

Comments

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.

Every term F(n,k) of F is a product of two Fibonacci numbers; indeed,

F(n,k)=F(n)*F(k+1) if k is even;

F(n,k)=F(n+1)*F(k) if k is odd.

antidiagonal sums: (1,2,6,12,...), A054454

diagonal (1,2,6,15,...), A001654

diagonal (1,3,9,24,...), A064831

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

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

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

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

principal submatrix sums, A202462

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..70.

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

Formula

Matrix product P*Q, where P, Q are the lower and upper triangular Fibonacci matrices, A202451 and A202452.

Example

Northwest corner:
1...1....2....3....5....8....13
1...2....3....5....8...13....21
2...3....6....9...15...24....39
3...5....9...15...24...39....63
5...8...15...24...40...64...104

Mathematica

n = 12;
Q = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[Fibonacci[k], {k, 1, n}]];
P = Transpose[Q]; F = P.Q;
Flatten[Table[P[[i]][[k + 1 - i]], {k, 1, n}, {i, 1, k}]] (* A202452 as a sequence *)
Flatten[Table[Q[[i]][[k + 1 - i]], {k, 1, n}, {i, 1, k}]] (* A202451 as a sequence *)
Flatten[Table[F[[i]][[k + 1 - i]], {k, 1, n}, {i, 1, k}]] (* A202453 as a sequence *)
TableForm[Q]  (* A202451, upper tri. Fibonacci array *)
TableForm[P]  (* A202452, lower tri. Fibonacci array *)
TableForm[F]  (* A202453, Fibonacci fusion array *)
TableForm[FactorInteger[F]]

Crossrefs

Cf. A000045, A202451, A202452, A202503 (Fibonacci fission array).

Sequence in context: A237050 A235130 A131410 * A259529 A196052 A080773

Adjacent sequences: A202450 A202451 A202452 * A202454 A202455 A202456

Keyword

nonn,tabl

Author

Clark Kimberling, Dec 19 2011

Status

approved