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

Author

Ralf Stephan, Jan 06 2005

Keywords

• nonn
• tabl

Examples

			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, ...
		

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

Crossrefs

Equals A000045(A003991(k, n)).

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

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

Programs

Magma

/* As triangle */ [[Fibonacci(k*(n-k+1)): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Jul 04 2019

Mathematica

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

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

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