A165357 Left-justified Wythoff Array.

1, 0, 2, 1, 1, 2, 1, 3, 0, 3, 2, 4, 2, 0, 4, 3, 7, 2, 3, 0, 3, 5, 11, 4, 3, 4, 1, 4, 8, 18, 6, 6, 4, 4, 1, 3, 13, 29, 10, 9, 8, 5, 5, 2, 4, 21, 47, 16, 15, 12, 9, 6, 5, 2, 5, 34, 76, 26, 24, 20, 14, 11, 7, 6, 2, 4, 55, 123, 42, 39, 32, 23, 17, 12, 8, 7, 3, 5, 89, 199, 68, 63, 52, 37, 28, 19, 14, 9

Offset

1,3

Comments

1. Every row satisfies the Fibonacci recurrence: x(n)=x(n-1)+x(n-2).

2. Row n of the Wythoff Array (A035513) is a tail of row n of A165357.

3. Every (a,b) having a>b>=0 occurs exactly once.

4. Every (c,d) having 0<=c<=d occurs exactly once.

References

C. Kimberling, "Ordering the set of all positive Fibonacci sequences," in Applications of Fibonacci Numbers, vol. 5, Proceedings of the Fifth" International Conference on Fibonacci Numbers and Their Applications, Kluwer, 1993, pages 405-416.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1 <= n <= 150, flattened).

Eric Duchêne, Aviezri S. Fraenkel, Vladimir Gurvich, Nhan Bao Ho, Clark Kimberling, Urban Larsson, Wythoff Visions, Games of No Chance 5 (2017) Vol. 70, See p. 65.

N. J. A. Sloane, Classic Sequences.

Formula

Row n is obtained from row n of the Wythoff array (A035513) by applying the reverse Fibonacci recurrence until reaching a pair (a,b) having a>b>=0.

Example

Northwest corner:
1 0 1 1 2 3
2 1 3 4 7 11
2 0 2 2 4 6
3 0 3 3 6 9
4 0 4 4 8 12
3 1 4 5 9 14
Row 6 of the Wythoff array is (14,23,37,60,...). Reverse recurrence gives
9=23-14, 5=14-9, 4=9-5, 1=5-4, 3=4-1, so that row 6 of A165357 is
(3,1,4,5,9,14,23,37,60,...).

Mathematica

w[n_, k_] := Fibonacci[k + 1] Floor[n GoldenRatio] + (n - 1) Fibonacci[k]; a[n_, k_] := w[n, Module[{z = 0}, ((While[w[#1, z] <= w[#1, z + 1], z--]; z - 1) &)[n] + k]]; Flatten[Table[a[n, k - n + 1], {k, 15}, {n, k}]] (* Peter J. C. Moses, Apr 11 2013 *)

Crossrefs

Cf. A000045, A165359, A165360.

Sequence in context: A319093 A228726 A346175 * A346741 A358105 A210961

Adjacent sequences: A165354 A165355 A165356 * A165358 A165359 A165360

Keyword

nonn,tabl

Author

Clark Kimberling, Sep 16 2009

Status

approved