A102310 Square array read by antidiagonals: Fibonacci(k*n).

1, 1, 1, 2, 3, 2, 3, 8, 8, 3, 5, 21, 34, 21, 5, 8, 55, 144, 144, 55, 8, 13, 144, 610, 987, 610, 144, 13, 21, 377, 2584, 6765, 6765, 2584, 377, 21, 34, 987, 10946, 46368, 75025, 46368, 10946, 987, 34, 55, 2584, 46368, 317811, 832040, 832040, 317811, 46368, 2584, 55

Offset

1,4

References

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. 2nd Edition. Addison-Wesley, Reading, MA, 1994, p. 294.

Links

Freddy Barrera, Antidiagonals n = 1..50, flattened

Formula

For prime p, the formula holds: Fibonacci(k*p) = Fibonacci(p) * Sum_{i=0..floor((k-1)/2)} C(k-i-1, i)*(-1)^(i*p+i)*Lucas(p)^(k-2i-1).

A(n, k) = F((n-1)*k)*F(k+1) + F((n-1)*k-1)*F(k), where F(n) = A000045(n). - Freddy Barrera, Jun 24 2019

Example

1,  1,   2,    3,     5, ...
1,  3,   8,   21,    55, ...
2,  8,  34,  144,   610, ...
3, 21, 144,  987,  6765, ...
5, 55, 610, 6765, 75025, ...

Mathematica

Table[Fibonacci[k*(n-k+1)], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 10 2017 *)

Prog

(Sage)
F = fibonacci # A000045
def A(n, k):
    return F((n-1)*k)*F(k+1) + F((n-1)*k - 1)*F(k)
[A(n, k) for d in (1..10) for n, k in zip((d..1, step=-1), (1..d))] # Freddy Barrera, Jun 24 2019
(Magma) /* As triangle */ [[Fibonacci(k*(n-k+1)): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Jul 04 2019

Crossrefs

Equals A000045(A003991(k, n)).

Columns include A000045, A001906, A014445, A033888, A102312.

Main diagonal is in A054783. Antidiagonal sums are in A102311.

Sequence in context: A341653 A085216 A300663 * A151546 A117936 A264766

Adjacent sequences: A102307 A102308 A102309 * A102311 A102312 A102313

Keyword

nonn,tabl

Author

Ralf Stephan, Jan 06 2005

Status

approved