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A257941 Lexicographically earliest sequence of positive integers such that the terms and their absolute first differences are all distinct and no term is the sum of two distinct earlier terms.

%I A257941 #20 Feb 16 2025 08:33:25
%S A257941 1,3,7,12,18,26,9,20,34,24,39,55,22,45,66,28,47,72,85,49,76,108,68,99,
%T A257941 53,82,112,70,114,149,74,122,172,93,145,203,101,160,95,162,216,118,
%U A257941 187,224,141,214,143,235,139,195,281,164,241,329,166,260,170,283,168
%N A257941 Lexicographically earliest sequence of positive integers such that the terms and their absolute first differences are all distinct and no term is the sum of two distinct earlier terms.
%C A257941 The sequence of absolute first differences begins: 2, 4, 5, 6, 8, 17, 11, 14, 10, 15, 16, 33, 23, 21, 38, 19, 25, 13, 36, 27, 32, 40, ... .
%C A257941 The sequence is 0-additive.
%H A257941 Alois P. Heinz, <a href="/A257941/b257941.txt">Table of n, a(n) for n = 1..10000</a>
%H A257941 E. Angelini et al., <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2015-May/014848.html">0-additive and first differences</a> and follow-up messages on the SeqFan list, May 13 2015
%H A257941 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/s-AdditiveSequence.html">s-Additive Sequence</a>
%p A257941 s:= proc() false end: b:= proc() false end:
%p A257941 a:= proc(n) option remember; local i, k;
%p A257941       if n=1 then b(1):= true; 1
%p A257941     else for k while b(k) or s(k) or
%p A257941          (t-> b(t) or t=k)(abs(a(n-1)-k)) do od;
%p A257941          for i to n-1 do s(a(i)+k):= true od;
%p A257941          b(k), b(abs(a(n-1)-k)):= true$2; k
%p A257941       fi
%p A257941     end:
%p A257941 seq(a(n), n=1..101);
%t A257941 s[_] = False; b[_] = False;
%t A257941 a[n_] := a[n] = Module[{i, k}, If[n == 1, b[1] = True; 1, For[k = 1, b[k] || s[k] || Function[t, b[t] || t == k][Abs[a[n-1]-k]], k++]; For[i = 1, i <= n-1, i++, s[a[i]+k] = True]; {b[k], b[Abs[a[n-1]-k]]} = {True, True}; k]];
%t A257941 Array[a, 101] (* _Jean-François Alcover_, Oct 28 2020, after Maple *)
%Y A257941 Cf. A005228, A030124, A033627, A095115, A140778, A257944.
%K A257941 nonn,look
%O A257941 1,2
%A A257941 _Eric Angelini_ and _Alois P. Heinz_, May 13 2015