

A202453


Fibonacci selffusion matrix, by antidiagonals.


65



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
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OFFSET

1,4


COMMENTS

The Fibonacci selffusion 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}.


REFERENCES

C. Kimberling, Fusion, Fission, and Factors, Fib. Q., 52 (2014), 195202.


LINKS

Table of n, a(n) for n=1..70.


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



