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A165357
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Left-justified Wythoff Array.
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6
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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
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OFFSET
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1,3
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COMMENTS
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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.
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REFERENCES
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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.
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LINKS
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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.
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FORMULA
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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.
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EXAMPLE
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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,...).
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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