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A257706 Sequence (a(n)) generated by Rule 1 (in Comments) with a(1) = 0 and d(1) = 1.

%I A257706 #9 May 14 2015 11:59:06
%S A257706 0,2,1,4,8,6,3,9,5,10,17,12,20,14,7,16,26,18,29,19,31,22,11,24,38,25,
%T A257706 13,28,44,30,15,32,50,34,53,36,56,37,58,40,62,42,21,45,23,46,71,48,74,
%U A257706 49,76,52,80,54,27,57,86,55,87,59,90,61,94,64,98,66,33
%N A257706 Sequence (a(n)) generated by Rule 1 (in Comments) with a(1) = 0 and d(1) = 1.
%C A257706 Rule 1 follows.  For k >= 1, let  A(k) = {a(1),..., a(k)} and D(k) = {d(1),..., d(k)}.  Begin with k = 1 and nonnegative integers a(1) and d(1).
%C A257706 Step 1:   If there is an integer h such that 1 - a(k) < h < 0 and h is not in D(k) and a(k) + h is not in A(k), let d(k+1) be the greatest such h, let a(k+1) = a(k) + h, replace k by k + 1, and repeat Step 1; otherwise do Step 2.
%C A257706 Step 2:  Let h be the least positive integer not in D(k) such that a(k) + h is not in A(k).  Let a(k+1) = a(k) + h and d(k+1) = h.  Replace k by k+1 and do Step 1.
%C A257706 Conjecture:  if a(1) is an nonnegative integer and d(1) is an integer, then (a(n)) is a permutation of the nonnegative integers (if a(1) = 0) or a permutation of the positive integers (if a(1) > 0).  Moreover, (d(n)) is a permutation of the integers if d(1) = 0, or of the nonzero integers if d(1) > 0.
%C A257706 See A257705 for a guide to related sequences.
%H A257706 Clark Kimberling, <a href="/A257706/b257706.txt">Table of n, a(n) for n = 1..1000</a>
%F A257706 a(n+1) - a(n) = d(n+1) = A131389(n+1) for n >= 1.
%e A257706 a(1) = 0, d(1) = 1;
%e A257706 a(2) = 2, d(2) = 2;
%e A257706 a(3) = 1, d(3) = -1;
%e A257706 a(4) = 4, d(4) = 3;
%e A257706 (The sequence d differs from A131389 only in the first 13 terms.)
%t A257706 a[1] = 0; d[1] = 1; k = 1; z = 10000; zz = 120;
%t A257706 A[k_] := Table[a[i], {i, 1, k}]; diff[k_] := Table[d[i], {i, 1, k}];
%t A257706 c[k_] := Complement[Range[-z, z], diff[k]];
%t A257706 T[k_] := -a[k] + Complement[Range[z], A[k]];
%t A257706 s[k_] := Intersection[Range[-a[k], -1], c[k], T[k]];
%t A257706 Table[If[Length[s[k]] == 0, {h = Min[Intersection[c[k], T[k]]], a[k + 1] = a[k] + h, d[k + 1] = h, k = k + 1}, {h = Max[s[k]], a[k + 1] = a[k] + h, d[k + 1] = h, k = k + 1}], {i, 1, zz}];
%t A257706 u = Table[a[k], {k, 1, zz}]  (* A257706 *)
%t A257706 Table[d[k], {k, 1, zz}]  (* A131389 shifted *)
%Y A257706 Cf. A131388, A257705, A081145, A257883, A175498.
%K A257706 nonn,easy
%O A257706 1,2
%A A257706 _Clark Kimberling_, May 12 2015