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A257705 Sequence (a(n)) generated by Rule 1 (in Comments) with a(1) = 0 and d(1) = 0.

%I A257705 #9 May 14 2015 11:58:54
%S A257705 0,1,3,2,5,9,7,4,10,6,11,18,13,21,15,8,17,27,19,30,20,32,23,12,25,39,
%T A257705 26,14,29,45,31,16,33,51,35,54,37,57,38,59,41,63,43,22,46,24,47,72,49,
%U A257705 75,50,77,53,81,55,28,58,87,56,88,60,91,62,95,65,99,67
%N A257705 Sequence (a(n)) generated by Rule 1 (in Comments) with a(1) = 0 and d(1) = 0.
%C A257705 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 A257705 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 A257705 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 A257705 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 A257705 Guide to related sequences:
%C A257705 a(1)  d(1)      (a(n))             (d(n))
%C A257705 0       0      A257705      A131389 except for initial terms
%C A257705 0       1      A257706      A131389 except for initial terms
%C A257705 0       2      A257876      A131389 except for initial terms
%C A257705 0       3      A257877      A257915
%C A257705 1       0      A131388      A131389
%C A257705 1       1      A257878      A131389 except for initial terms
%C A257705 2       0      A257879      A257880
%C A257705 2       1      A257881      A257880 except for initial terms
%C A257705 2       2      A257882      A257918
%H A257705 Clark Kimberling, <a href="/A257705/b257705.txt">Table of n, a(n) for n = 1..1000</a>
%F A257705 a(k+1) - a(k) = d(k+1) for k >= 1.
%F A257705 Also, a(k) = A131388(n)-1.
%e A257705 a(2) = a(1) + d(2) = 0 + 1 = 1;
%e A257705 a(3) = a(2) + d(3) = 1 + 2 = 3;
%e A257705 a(4) = a(3) + d(4) = 3 + (-1) = 2.
%t A257705 a[1] = 0; d[1] = 0; k = 1; z = 10000; zz = 120;
%t A257705 A[k_] := Table[a[i], {i, 1, k}]; diff[k_] := Table[d[i], {i, 1, k}];
%t A257705 c[k_] := Complement[Range[-z, z], diff[k]];
%t A257705 T[k_] := -a[k] + Complement[Range[z], A[k]];
%t A257705 s[k_] := Intersection[Range[-a[k], -1], c[k], T[k]];
%t A257705 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 A257705 u = Table[a[k], {k, 1, zz}] (* A257705 *)
%t A257705 Table[d[k], {k, 1, zz}]     (* A131389 *)
%Y A257705 Cf. A131388, A081145, A257883, A175498.
%K A257705 nonn,easy
%O A257705 1,3
%A A257705 _Clark Kimberling_, May 12 2015