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
Keywords
Examples
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, ... ...
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.
Crossrefs
Programs
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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
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
Extensions
a(42)-a(67) from Pontus von Brömssen, Jan 15 2025