%I #52 Aug 06 2024 15:05:44
%S 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,
%T 74,82,84,90,93,100,107,123,139,155,157,159,160,171,184,198,230,259,
%U 263,280,288,293,296,303,343,392,442,474,496,500,506,507,512,532,573,622,725,781,815,881,883
%N The two-dimensional Zeckendorf lattice T(m,n) (m>=1, n>=1) read by downward antidiagonals.
%C This is called the "Simple Zeckendorf diagonal sequence in two dimensions". - _N. J. A. Sloane_, Dec 11 2020
%C The array T(i,j) is constructed as follows, by filling in the array by downward antidiagonals, in this order:
%C 1, 2, 4, 7, ...
%C 3, 5, 8, ...
%C 6, 9, ...
%C 10, ...
%C Start with T(1,1) = 1.
%C 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.
%C 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.
%C 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.
%D 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.
%H Rémy Sigrist, <a href="/A335154/b335154.txt">Table of n, a(n) for n = 1..630</a> (first 35 rows flattened)
%H Neelima Borade, Dexter Cai, David Z. Chang, Bruce Fang, Alex Liang, Steven J. Miller, and Wanqiao Xu, <a href="https://arxiv.org/abs/1909.01935">Gaps of Summands of the Zeckendorf Lattice</a>, arXiv:1909.01935 [math.NT], 2019.
%H Chen, E., Chen, R., Guo, L., Jiang, C., Miller, S. J., Siktar, J. M., & Yu, P. (2018). <a href="https://arxiv.org/abs/1809.05829">Gaussian Behavior in Zeckendorf Decompositions From Lattices</a>, arXiv preprint arXiv:1809.05829. Also Fib. Q., 57:5 (2019), 201-212.
%H Fang, E., Jenkins, J., Lee, Z., Li, D., Lu, E., Miller, S. J., ... & Siktar, J. (2019). <a href="https://arxiv.org/abs/1906.10645">Central Limit Theorems for Compound Paths on the 2-Dimensional Lattice</a>, arXiv preprint arXiv:1906.10645. Also Fib. Q., 58:1 (2020), 208-225.
%H Rémy Sigrist, <a href="/A335154/a335154.gp.txt">PARI program for A335154</a>
%e The array begins:
%e 1, 2, 4, 8, 16, 29, 54, 90, 159, ...
%e 3, 5, 9, 17, 30, 56, 93, 160, ...
%e 7, 12, 20, 33, 59, 100, 171, ...
%e 14, 24, 40, 66, 107, 184, ...
%e 28, 48, 74, 123, 198, ...
%e 50, 82, 139, 230, ...
%e 84, 155, 259, ...
%e 157, 263, ...
%e 280, ...
%e ...
%e The first few antidiagonals are:
%e 1;
%e 2, 3;
%e 4, 5, 7;
%e 8, 9, 12, 14;
%e 16, 17, 20, 24, 28;
%e 29, 30, 33, 40, 48, 50;
%e ...
%e In Chen et al. (2019) the array appears as follows:
%e ...
%e 280, ...
%e 157, 263, ...
%e 84, 155, 259, ...
%e 50, 82, 139, 230, ...
%e 28, 48, 74, 123, 198, ...
%e 14, 24, 40, 66, 107, 184, ...
%e 7, 12, 20, 33, 59, 100, 171, ...
%e 3, 5, 9, 17, 30, 56, 93, 160, ...
%e 1, 2, 4, 8, 16, 29, 54, 90, 154, ...
%o (PARI) \\ See Links section.
%Y Cf. A339574.
%K nonn,tabl
%O 1,2
%A _N. J. A. Sloane_, Jun 02 2020
%E More terms from _Rémy Sigrist_, Jun 10 2020