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A335154
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The two-dimensional Zeckendorf lattice T(m,n) (m>=1, n>=1) read by downward antidiagonals.
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2
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1, 2, 3, 4, 5, 7, 8, 9, 12, 14, 16, 17, 20, 24, 28, 29, 30, 33, 40, 48, 50, 54, 56, 59, 66, 74, 82, 84, 90, 93, 100, 107, 123, 139, 155, 157, 159, 160, 171, 184, 198, 230, 259, 263, 280, 288, 293, 296, 303, 343, 392, 442, 474, 496, 500, 506, 507, 512, 532, 573, 622, 725, 781, 815, 881, 883
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OFFSET
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1,2
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COMMENTS
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This is called the "Simple Zeckendorf diagonal sequence in two dimensions". - N. J. A. Sloane, Dec 11 2020
The array T(i,j) is constructed as follows, by filling in the array by downward antidiagonals, in this order:
1, 2, 4, 7, ...
3, 5, 8, ...
6, 9, ...
10, ...
Start with T(1,1) = 1.
Look at the next empty square (on the succession of antidiagonal paths). Iterate through the natural numbers, starting at the last number that we accepted. When we consider a number k, if we can get k as a sum of existing numbers that are on a path in the array starting at any filled square and always moves up and to the left, skip k, and consider k+1. Otherwise, enter k in the square.
For example, T(3,1) is not 6 because we can get 6 as the sum of 5 and 1. So T(3,1) = 7.
T(3,2) is not 10, because we can get 10 as 9+1, and it not 11 because of 9+2. So T(3,2) = 12.
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REFERENCES
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Borade, N., Cai, D., Chang, D. Z., Fang, B., Liang, A., Miller, S. J., & Xu, W. (2019). Gaps of Summands of the Zeckendorf Lattice. Fib. Q., 58:2 (2020), 143-156. See the array y_{i,j}, although in the present entry the order of the rows has been reversed.
Chen, E., Chen, R., Guo, L., Jiang, C., Miller, S. J., Siktar, J. M., & Yu, P. (2018). Gaussian Behavior in Zeckendorf Decompositions From Lattices. arXiv preprint arXiv:1809.05829. Also Fib. Q., 57:5 (2019), 201-212.
Fang, E., Jenkins, J., Lee, Z., Li, D., Lu, E., Miller, S. J., ... & Siktar, J. (2019). Central Limit Theorems for Compound Paths on the 2-Dimensional Lattice. arXiv preprint arXiv:1906.10645. Also Fib. Q., 58:1 (2020), 208-225.
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LINKS
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EXAMPLE
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The array begins:
1, 2, 4, 8, 16, 29, 54, 90, 159, ...
3, 5, 9, 17, 30, 56, 93, 160, ...
7, 12, 20, 33, 59, 100, 171, ...
14, 24, 40, 66, 107, 184, ...
28, 48, 74, 123, 198, ...
50, 82, 139, 230, ...
84, 155, 259, ...
157, 263, ...
280, ...
...
The first few antidiagonals are:
1;
2, 3;
4, 5, 7;
8, 9, 12, 14;
16, 17, 20, 24, 28;
29, 30, 33, 40, 48, 50;
...
In Chen et al. (2019) the array appears as follows:
...
280, ...
157, 263, ...
84, 155, 259, ...
50, 82, 139, 230, ...
28, 48, 74, 123, 198, ...
14, 24, 40, 66, 107, 184, ...
7, 12, 20, 33, 59, 100, 171, ...
3, 5, 9, 17, 30, 56, 93, 160, ...
1, 2, 4, 8, 16, 29, 54, 90, 154, ...
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PROG
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(PARI) See Links section.
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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