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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).

%I A368050 #33 Jan 19 2025 14:26:22
%S A368050 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,
%T A368050 32,33,21,36,37,38,24,41,42,43,27,28,47,48,30,31,52,53,33,34,57,58,59,
%U A368050 37,62,63,64,40,41,68,69,43,44,73,74,46,47,78,79,80,50,83
%N 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).
%H A368050 Pontus von Brömssen, <a href="/A368050/b368050.txt">Table of n, a(n) for n = 1..10000</a>
%H A368050 Jeffrey Shallit, <a href="/A368050/a368050.pdf">Automaton that decides whether a(n) >= n</a>
%H A368050 Jeffrey Shallit, <a href="https://arxiv.org/abs/2501.08823">The Hurt-Sada Array and Zeckendorf Representations</a>, arXiv:2501.08823 [math.NT], 2025. See pp. 2, 12.
%F A368050 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
%e A368050 The array begins:
%e A368050   1,  2,  3,  4,  5,  6,  7,  8,  9,  10,  11,  12,  13,  ...
%e A368050   2,  1,  3,  4,  5,  6,  7,  8,  9,  10,  11,  12,  13,  ...
%e A368050   1,  3,  2,  4,  5,  6,  7,  8,  9,  10,  11,  12,  13,  ...
%e A368050   1,  2,  4,  5,  3,  6,  7,  8,  9,  10,  11,  12,  13,  ...
%e A368050   1,  2,  5,  3,  6,  7,  4,  8,  9,  10,  11,  12,  13,  ...
%e A368050   1,  2,  3,  6,  7,  4,  8,  5,  9,  10,  11,  12,  13,  ...
%e A368050   1,  2,  3,  7,  4,  8,  5,  9, 10,   6,  11,  12,  13,  ...
%e A368050   1,  2,  3,  4,  8,  5,  9, 10,  6,  11,   7,  12,  13,  ...
%e A368050   1,  2,  3,  4,  5,  9, 10,  6, 11,   7,  12,  13,   8,  ...
%e A368050   1,  2,  3,  4,  5, 10,  6, 11,  7,  12,  13,   8,  14,  ...
%e A368050   1,  2,  3,  4,  5,  6, 11,  7, 12,  13,   8,  14,  15,  ...
%e A368050   1,  2,  3,  4,  5,  6,  7, 12, 13,   8,  14,  15,   9,  ...
%e A368050   1,  2,  3,  4,  5,  6,  7, 13,  8,  14,  15,   9,  16,  ...
%e A368050   ...
%o A368050 (Python)
%o A368050 from itertools import count
%o A368050 def A368050_generator():
%o A368050     x = [1]
%o A368050     for n in count(1):
%o A368050         yield x[n-1]
%o A368050         i = x.index(n)
%o A368050         if len(x) <= i+n: x.extend(range(len(x)+1,i+n+2))
%o A368050         x[i:i+n] = x[i+1:i+n+1]
%o A368050         x[i+n] = n # _Pontus von Brömssen_, Jan 15 2025
%Y A368050 Other parts of the array: A379739 (subdiagonal) and A367634 (descending antidiagonals).
%Y A368050 Cf. A000027, A026272.
%K A368050 nonn
%O A368050 1,3
%A A368050 _Wesley Ivan Hurt_, Dec 09 2023
%E A368050 a(42)-a(67) from _Pontus von Brömssen_, Jan 15 2025