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A257876 Sequence (a(n)) generated by Rule 1 (in Comments) with a(1) = 0 and d(1) = 2.

%I A257876 #10 May 14 2015 12:52:15
%S A257876 0,1,4,3,7,5,2,8,13,9,16,11,19,12,6,15,25,17,28,18,30,21,10,23,37,24,
%T A257876 39,27,43,29,14,31,49,33,52,35,55,36,57,34,56,38,61,41,20,44,22,47,73,
%U A257876 48,75,51,79,53,26,58,87,59,89,60,91,54,88,50,83,42,77
%N A257876 Sequence (a(n)) generated by Rule 1 (in Comments) with a(1) = 0 and d(1) = 2.
%C A257876 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 A257876 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 A257876 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 A257876 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 A257876 See A257705 for a guide to related sequences.
%H A257876 Clark Kimberling, <a href="/A257876/b257876.txt">Table of n, a(n) for n = 1..1000</a>
%F A257876 a(k+1) - a(k) = d(k+1) for k >= 1.
%e A257876 a(1) = 0, d(1) = 2;
%e A257876 a(2) = 1, d(2) = 1;
%e A257876 a(3) = 3, d(3) = 3;
%e A257876 a(4) = 4, d(4) = -1.
%e A257876 The first terms of (d(n)) are (2,1,3,-1,4,-2,-3,6,5,...), which differs from A131389 only in initial terms.
%t A257876 a[1] = 0; d[1] = 2; k = 1; z = 10000; zz = 120;
%t A257876 A[k_] := Table[a[i], {i, 1, k}]; diff[k_] := Table[d[i], {i, 1, k}];
%t A257876 c[k_] := Complement[Range[-z, z], diff[k]];
%t A257876 T[k_] := -a[k] + Complement[Range[z], A[k]];
%t A257876 s[k_] := Intersection[Range[-a[k], -1], c[k], T[k]];
%t A257876 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 A257876 u = Table[a[k], {k, 1, zz}]  (* A257876 *)
%t A257876 Table[d[k], {k, 1, zz}]      (* A131389 essentially *)
%Y A257876 Cf. A131388, A257705, A081145, A257883, A175498.
%K A257876 nonn,easy
%O A257876 1,3
%A A257876 _Clark Kimberling_, May 12 2015