A257941 - OEIS

%I #19 Dec 23 2024 14:53:44

%S 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 53,82,112,70,114,149,74,122,172,93,145,203,101,160,95,162,216,118,

%U 187,224,141,214,143,235,139,195,281,164,241,329,166,260,170,283,168

%N 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 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 The sequence is 0-additive.

%H Alois P. Heinz, <a href="/A257941/b257941.txt">Table of n, a(n) for n = 1..10000</a>

%H 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/s-AdditiveSequence.html">s-Additive Sequence</a>

%p s:= proc() false end: b:= proc() false end:

%p a:= proc(n) option remember; local i, k;

%p       if n=1 then b(1):= true; 1

%p     else for k while b(k) or s(k) or

%p          (t-> b(t) or t=k)(abs(a(n-1)-k)) do od;

%p          for i to n-1 do s(a(i)+k):= true od;

%p          b(k), b(abs(a(n-1)-k)):= true$2; k

%p       fi

%p     end:

%p seq(a(n), n=1..101);

%t s[_] = False; b[_] = False;

%t 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 Array[a, 101] (* _Jean-François Alcover_, Oct 28 2020, after Maple *)

%Y Cf. A005228, A030124, A033627, A095115, A140778, A257944.

%K nonn,look

%O 1,2

%A _Eric Angelini_ and _Alois P. Heinz_, May 13 2015