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A131388 Sequence (a(n)) generated by Rule 1 (in Comments) with a(1) = 1 and d(1) = 0.

%I A131388 #12 May 14 2015 12:51:58
%S A131388 1,2,4,3,6,10,8,5,11,7,12,19,14,22,16,9,18,28,20,31,21,33,24,13,26,40,
%T A131388 27,15,30,46,32,17,34,52,36,55,38,58,39,60,42,64,44,23,47,25,48,73,50,
%U A131388 76,51,78,54,82,56,29,59,88,57,89,61,92,63,96,66,100,68,35,70,106,72,37
%N A131388 Sequence (a(n)) generated by Rule 1 (in Comments) with a(1) = 1 and d(1) = 0.
%C A131388 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 A131388 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 A131388 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 A131388 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 A131388 See A257705 for a guide to related sequences.
%H A131388 Clark Kimberling, <a href="/A131388/b131388.txt">Table of n, a(n) for n = 1..1000</a>
%F A131388 a(k+1) - a(k) = d(k+1) for k >= 1.
%e A131388 a(2)=1+1, a(3)=a(2)+2, a(4)=a(3)+(-1), a(5)=a(4)+3, a(6)=a(5)+4.
%t A131388 (*Program 1 *)
%t A131388 {a, f} = {{1}, {0}}; Do[tmp = {#, # - Last[a]} &[Max[Complement[#, Intersection[a, #]] &[Last[a] + Complement[#, Intersection[f, #]] &[Range[2 - Last[a], -1]]]]];
%t A131388 If[! IntegerQ[tmp[[1]]], tmp = {Last[a] + #, #} &[NestWhile[# + 1 &, 1, ! (! MemberQ[f,#] && ! MemberQ[a, Last[a] + #]) &]]];
%t A131388 AppendTo[a, tmp[[1]]]; AppendTo[f, tmp[[2]]], {400}];
%t A131388 {a, f} (*{A131388, A131389}; _Peter J. C. Moses_, May 10 2015*)
%t A131388 (*Program 2 *)
%t A131388 a[1] = 1; d[1] = 0; k = 1; z = 10000; zz = 120;
%t A131388 A[k_] := Table[a[i], {i, 1, k}]; diff[k_] := Table[d[i], {i, 1, k}];
%t A131388 c[k_] := Complement[Range[-z, z], diff[k]];
%t A131388 T[k_] := -a[k] + Complement[Range[z], A[k]];
%t A131388 s[k_] := Intersection[Range[-a[k], -1], c[k], T[k]];
%t A131388 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 A131388 u = Table[a[k], {k, 1, zz}] (* A131388 *)
%t A131388 Table[d[k], {k, 1, zz}]     (* A131389 *)
%Y A131388 Cf. A131389, A131390, A131391, A131392, A131393, A131394, A131395, A131396, A131397, A257705, A175498.
%K A131388 nonn
%O A131388 1,2
%A A131388 _Clark Kimberling_, Jul 05 2007
%E A131388 Revised by _Clark Kimberling_, May 12 2015