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A257877 Sequence (a(n)) generated by Rule 1 (in Comments) with a(1) = 0 and d(1) = 3.

%I A257877 #9 May 14 2015 12:52:23
%S A257877 0,1,3,2,6,4,9,5,11,8,15,10,18,12,21,14,24,16,7,19,30,20,33,22,36,23,
%T A257877 38,26,42,28,13,31,48,32,51,34,54,35,17,39,60,40,63,41,65,44,69,46,72,
%U A257877 47,74,50,78,52,25,55,27,56,87,58,90,59,29,62,96,64,99
%N A257877 Sequence (a(n)) generated by Rule 1 (in Comments) with a(1) = 0 and d(1) = 3.
%C A257877 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 A257877 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 A257877 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 A257877 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 A257877 See A257705 for a guide to related sequences.
%H A257877 Clark Kimberling, <a href="/A257877/b257877.txt">Table of n, a(n) for n = 1..1000</a>
%F A257877 a(k+1) - a(k) = d(k+1) for k >= 1.
%e A257877 a(1) = 0, d(1) = 3;
%e A257877 a(2) = 1, d(2) = 1;
%e A257877 a(3) = 3, d(3) = 2;
%e A257877 a(4) = 2, d(4) = -1.
%t A257877 a[1] = 0; d[1] = 3; k = 1; z = 10000; zz = 120;
%t A257877 A[k_] := Table[a[i], {i, 1, k}]; diff[k_] := Table[d[i], {i, 1, k}];
%t A257877 c[k_] := Complement[Range[-z, z], diff[k]];
%t A257877 T[k_] := -a[k] + Complement[Range[z], A[k]];
%t A257877 s[k_] := Intersection[Range[-a[k], -1], c[k], T[k]];
%t A257877 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 A257877 u = Table[a[k], {k, 1, zz}]  (* A257877 *)
%t A257877 Table[d[k], {k, 1, zz}]  (* A257915 *)
%Y A257877 Cf. A131388, A257915, A257705, A081145, A257883, A175498.
%K A257877 nonn,easy
%O A257877 1,3
%A A257877 _Clark Kimberling_, May 12 2015