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A356080 Variation on Recamán's sequence (A005132) that is intended to be a permutation of the nonnegative integers, essentially as envisaged by the original comments in A078943. See comments below for details.

%I A356080 #34 Aug 07 2024 15:11:14
%S A356080 0,1,3,6,2,7,13,20,28,37,27,16,4,17,31,46,30,47,29,48,68,89,111,134,
%T A356080 158,133,159,132,160,131,101,70,38,5,39,74,110,147,109,148,108,149,
%U A356080 107,150,106,151,105,58,10,59,9,60,8,61,115,170,226,283,341,400,460,399,337,274,210,145,79,12,80,11
%N A356080 Variation on Recamán's sequence (A005132) that is intended to be a permutation of the nonnegative integers, essentially as envisaged by the original comments in A078943. See comments below for details.
%C A356080 a(1) = 0; for n >= 1, a(n+1) = a(n)-n or a(n)+n; and no two terms are equal.
%C A356080 Subject to the pairwise constraints above, the sequence describes a path between (nodes labeled with) nonnegative integers that is extended in stages so that the extended path ends with the least missing number from the unextended path.
%C A356080 We denote the number at the end of the unextended path as k, the path from 0 to k as P_k and the least number absent from P_k as m_k. We call a path that starts at k an extension route if it extends P_k and satisfies the pairwise constraints. [clarification added, _Peter Munn_, Aug 07 2024]
%C A356080 An extension route R_k from P_k is suitable if (1) R_k ends in m_k and (2) R_k is the start of a "look-ahead" extension route (R_k + R') that ends in a record number (i.e., greater than the largest in P_k + R_k). If there is no suitable R_k the sequence finishes at k. Otherwise we denote the lexicographically earliest shortest such R_k as E_m_k, and extend P_k as P_m_k = P_k + E_m_k.
%C A356080 Note that any E_m_k ends in m_k (i.e., without any look-ahead R' being included) and so E_k ends in k and beware the sequence might end there. Nevertheless, once we determine that the sequence continues and includes m_k (perhaps because at least 1 suitable R_k has been found) 1 or more terms of E_m_k will be known, and this is the point in time we may include terms after k in the published sequence.
%C A356080 Conjecture (_Peter Munn_): the sequence is infinite and so a permutation of the nonnegative integers.
%H A356080 Peter Munn, <a href="/A356080/b356080.txt">Table of n, a(n) for n = 1..424</a>
%H A356080 Peter Munn and Ali Sada, <a href="/A356080/a356080.txt">Selection of relevant SeqFan posts</a>, July 2022.
%H A356080 Peter Munn, <a href="/A356080/a356080_1.txt">Initial validation program (LibreOffice Basic)</a>
%F A356080 |a(n) - a(n+1)| = n.
%F A356080 If a(n) = a(m) then n = m.
%e A356080 a(2) = a(1) + 1 = 1, since a(1) - 1 = -1 is a negative integer.
%e A356080 We now find the lexicographically earliest shortest route to the least missing number, 2. Any extension route has a(3) = a(2) + 2 = 3 since a(2) - 2 = -1 is a negative integer. Any extension route has a(4) = a(3) + 3 = 6, since a(3) - 3 = 0 is already in the sequence. So a(3) = 3, a(4) = 6, a(5) = a(4) - 4 = 2 is the only way to reach 2 by a(5); no shorter route exists. Lastly, we must check an onward route exists to a new record term (greater than 6). This is provided by a(6) = a(5) + 5 = 7, so we have determined a(3) = 3, a(4) = 6, a(5) = 2.
%o A356080 (LibreOffice Basic) See Links section.
%Y A356080 Cf. A005132, A078943.
%K A356080 nonn
%O A356080 1,3
%A A356080 _Peter Munn_, Jul 25 2022