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A257885 Sequence (a(n)) generated by Algorithm (in Comments) with a(1) = 0 and d(1) = 2.

%I A257885 #9 Jun 09 2015 14:21:52
%S A257885 0,1,4,2,6,3,8,7,13,5,12,20,9,18,11,21,15,10,22,33,14,27,17,31,16,32,
%T A257885 19,34,25,42,24,43,23,41,29,49,26,47,30,52,28,51,35,59,37,62,36,63,38,
%U A257885 64,50,46,74,39,68,40,70,101,44,76,45,78,48,82,53,88,54
%N A257885 Sequence (a(n)) generated by Algorithm (in Comments) with a(1) = 0 and d(1) = 2.
%C A257885 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 A257885 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 A257885 See A257883 for a guide to related sequences.
%H A257885 Clark Kimberling, <a href="/A257885/b257885.txt">Table of n, a(n) for n = 1..1000</a>
%F A257885 a(k+1) - a(k) = d(k+1) for k >= 1.
%e A257885 a(1) = 0, d(1) = 2;
%e A257885 a(2) = 1, d(2) = 1;
%e A257885 a(3) = 4, d(3) = 3;
%e A257885 a(4) = 2, d(4) = -2.
%t A257885 a[1] = 0; d[1] = 2; k = 1; z = 10000; zz = 120;
%t A257885 A[k_] := Table[a[i], {i, 1, k}]; diff[k_] := Table[d[i], {i, 1, k}];
%t A257885 c[k_] := Complement[Range[-z, z], diff[k]];
%t A257885 T[k_] := -a[k] + Complement[Range[z], A[k]]
%t A257885 Table[{h = Min[Intersection[c[k], T[k]]], a[k + 1] = a[k] + h,
%t A257885    d[k + 1] = h, k = k + 1}, {i, 1, zz}];
%t A257885 u = Table[a[k], {k, 1, zz}]  (* A257885 *)
%t A257885 Table[d[k], {k, 1, zz}] (* A257902 *)
%Y A257885 Cf. A257902, A257883, A257705.
%K A257885 nonn,easy
%O A257885 1,3
%A A257885 _Clark Kimberling_, May 13 2015