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A335154 The two-dimensional Zeckendorf lattice T(m,n) (m>=1, n>=1) read by downward antidiagonals. 2
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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.

REFERENCES

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.

LINKS

Rémy Sigrist, Table of n, a(n) for n = 1..630 (first 35 rows flattened)

Neelima Borade, Dexter Cai, David Z. Chang, Bruce Fang, Alex Liang, Steven J. Miller, and Wanqiao Xu, Gaps of Summands of the Zeckendorf Lattice, arXiv:1909.01935 [math.NT], 2019.

Rémy Sigrist, PARI program for A335154

EXAMPLE

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, ...

PROG

(PARI) See Links section.

CROSSREFS

Cf. A339574.

Sequence in context: A241480 A211656 A051204 * A278181 A232566 A192649

Adjacent sequences:  A335151 A335152 A335153 * A335155 A335156 A335157

KEYWORD

nonn,tabl,changed

AUTHOR

N. J. A. Sloane, Jun 02 2020

EXTENSIONS

More terms from Rémy Sigrist, Jun 10 2020

STATUS

approved

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Last modified January 23 17:33 EST 2022. Contains 350514 sequences. (Running on oeis4.)