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A257912 Sequence (d(n)) generated by Algorithm (in Comments) with a(1) = 2 and d(1) = 2.

%I A257912 #6 Jun 16 2015 13:54:02
%S A257912 2,-1,3,1,-2,4,5,-6,7,-5,6,-4,8,-9,10,-8,9,-3,11,-13,12,-11,13,-7,14,
%T A257912 -15,16,-14,15,-12,17,-19,18,-17,19,-10,20,-24,21,-20,22,-21,23,-22,
%U A257912 24,-18,25,-29,26,-16,27,-32,28,-27,29,-23,30,-31,32,-30,31,33
%N A257912 Sequence (d(n)) generated by Algorithm (in Comments) with a(1) = 2 and d(1) = 2.
%C A257912 Algorithm: 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). Let h be the least integer > -a(k) such that h is not in D(k) and 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 repeat inductively.
%C A257912 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 A257912 See A257883 for a guide to related sequences.
%H A257912 Clark Kimberling, <a href="/A257912/b257912.txt">Table of n, a(n) for n = 1..1000</a>
%t A257912 a[1] = 2; d[1] = 2; k = 1; z = 10000; zz = 120;
%t A257912 A[k_] := Table[a[i], {i, 1, k}]; diff[k_] := Table[d[i], {i, 1, k}];
%t A257912 c[k_] := Complement[Range[-z, z], diff[k]];
%t A257912 T[k_] := -a[k] + Complement[Range[z], A[k]];
%t A257912 Table[{h = Min[Intersection[c[k], T[k]]], a[k + 1] = a[k] + h, d[k + 1] = h, k = k + 1}, {i, 1, zz}];
%t A257912 Table[a[k], {k, 1, zz}]  (* A257911 *)
%t A257912 Table[d[k], {k, 1, zz}]  (* A257912 *)
%Y A257912 Cf. A257911, A257883.
%K A257912 easy,sign
%O A257912 1,1
%A A257912 _Clark Kimberling_, Jun 12 2015