A368050 Main diagonal of the array where row n=0 lists the natural numbers and each new row n=1,2,... is found by taking the number n in the previous row, and "leaping" it over the next n terms to its right, keeping the other numbers fixed (see example).

1, 1, 2, 5, 6, 4, 5, 10, 11, 12, 8, 15, 16, 17, 11, 20, 21, 22, 14, 15, 26, 27, 17, 18, 31, 32, 33, 21, 36, 37, 38, 24, 41, 42, 43, 27, 28, 47, 48, 30, 31, 52, 53, 33, 34, 57, 58, 59, 37, 62, 63, 64, 40, 41, 68, 69, 43, 44, 73, 74, 46, 47, 78, 79, 80, 50, 83

Offset

1,3

Links

Pontus von Brömssen, Table of n, a(n) for n = 1..10000

Jeffrey Shallit, Automaton that decides whether a(n) >= n

Jeffrey Shallit, The Hurt-Sada Array and Zeckendorf Representations, arXiv:2501.08823 [math.NT], 2025. See pp. 2, 12.

Formula

If a(n) >= n, then a(n) = floor((2*g-2)n + 1/2), where g = (1+sqrt(5))/2 is the golden ratio. If a(n) < n, then a(n) = floor((4-2*g)*n). There is a 6-state automaton (in the "links" section) that takes the Zeckendorf representation of n and accepts if and only if a(n) >= n. - Jeffrey Shallit, Jan 14 2025

Example

The array begins:
  1,  2,  3,  4,  5,  6,  7,  8,  9,  10,  11,  12,  13,  ...
  2,  1,  3,  4,  5,  6,  7,  8,  9,  10,  11,  12,  13,  ...
  1,  3,  2,  4,  5,  6,  7,  8,  9,  10,  11,  12,  13,  ...
  1,  2,  4,  5,  3,  6,  7,  8,  9,  10,  11,  12,  13,  ...
  1,  2,  5,  3,  6,  7,  4,  8,  9,  10,  11,  12,  13,  ...
  1,  2,  3,  6,  7,  4,  8,  5,  9,  10,  11,  12,  13,  ...
  1,  2,  3,  7,  4,  8,  5,  9, 10,   6,  11,  12,  13,  ...
  1,  2,  3,  4,  8,  5,  9, 10,  6,  11,   7,  12,  13,  ...
  1,  2,  3,  4,  5,  9, 10,  6, 11,   7,  12,  13,   8,  ...
  1,  2,  3,  4,  5, 10,  6, 11,  7,  12,  13,   8,  14,  ...
  1,  2,  3,  4,  5,  6, 11,  7, 12,  13,   8,  14,  15,  ...
  1,  2,  3,  4,  5,  6,  7, 12, 13,   8,  14,  15,   9,  ...
  1,  2,  3,  4,  5,  6,  7, 13,  8,  14,  15,   9,  16,  ...
  ...

Prog

(Python)
from itertools import count
def A368050_generator():
    x = [1]
    for n in count(1):
        yield x[n-1]
        i = x.index(n)
        if len(x) <= i+n: x.extend(range(len(x)+1, i+n+2))
        x[i:i+n] = x[i+1:i+n+1]
        x[i+n] = n # Pontus von Brömssen, Jan 15 2025

Crossrefs

Other parts of the array: A379739 (subdiagonal) and A367634 (descending antidiagonals).

Cf. A000027, A026272.

Sequence in context: A213736 A202343 A154946 * A343933 A262152 A016636

Adjacent sequences: A368047 A368048 A368049 * A368051 A368052 A368053

Keyword

nonn

Author

Wesley Ivan Hurt, Dec 09 2023

Extensions

a(42)-a(67) from Pontus von Brömssen, Jan 15 2025

Status

approved