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A257878 Sequence (a(n)) generated by Rule 1 (in Comments) with a(1) = 1 and d(1) = 1.

%I A257878 #12 May 14 2015 17:35:54
%S A257878 1,3,2,5,9,7,4,10,6,11,18,13,21,15,8,17,27,19,30,20,32,23,12,25,39,26,
%T A257878 14,29,45,31,16,33,51,35,54,37,57,38,59,41,63,43,22,46,24,47,72,49,75,
%U A257878 50,77,53,81,55,28,58,87,56,88,60,91,62,95,65,99,67,34
%N A257878 Sequence (a(n)) generated by Rule 1 (in Comments) with a(1) = 1 and d(1) = 1.
%C A257878 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 A257878 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 A257878 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 A257878 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 A257878 See A257705 for a guide to related sequences.
%C A257878 Considering the first 1000 elements of this sequence and A257705 it appears that this is the same as A257705 apart from an index shift. - _R. J. Mathar_, May 14 2015
%H A257878 Clark Kimberling, <a href="/A257878/b257878.txt">Table of n, a(n) for n = 1..1000</a>
%F A257878 a(k+1) - a(k) = d(k+1) for k >= 1.
%e A257878 a(1) = 1, d(1) = 1;
%e A257878 a(2) = 3, d(2) = 2;
%e A257878 a(3) = 2, d(3) = -1;
%e A257878 a(4) = 5, d(4) = -3.
%t A257878 a[1] = 1; d[1] = 1; k = 1; z = 10000; zz = 120;
%t A257878 A[k_] := Table[a[i], {i, 1, k}]; diff[k_] := Table[d[i], {i, 1, k}];
%t A257878 c[k_] := Complement[Range[-z, z], diff[k]];
%t A257878 T[k_] := -a[k] + Complement[Range[z], A[k]];
%t A257878 s[k_] := Intersection[Range[-a[k], -1], c[k], T[k]];
%t A257878 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 A257878 u = Table[a[k], {k, 1, zz}]  (* A257878 *)
%t A257878 Table[d[k], {k, 1, zz}]      (* A131389 essentially *)
%Y A257878 Cf. A131388, A131389, A257705, A081145, A257883, A175498.
%K A257878 nonn,easy
%O A257878 1,2
%A A257878 _Clark Kimberling_, May 12 2015