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A257903 Sequence (a(n)) generated by Algorithm (in Comments) with a(1) = 0 and d(1) = 3.

%I A257903 #9 Jun 11 2015 10:35:42
%S A257903 0,1,3,2,6,4,9,5,11,8,15,7,16,10,18,13,23,12,24,14,25,38,17,31,19,34,
%T A257903 20,36,21,39,22,41,28,45,26,46,30,51,27,49,29,52,43,67,32,57,35,61,33,
%U A257903 60,37,65,40,69,42,72,54,47,78,44,76,50,83,53,87,48,84
%N A257903 Sequence (a(n)) generated by Algorithm (in Comments) with a(1) = 0 and d(1) = 3.
%C A257903 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 A257903 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 A257903 See A257883 for a guide to related sequences.
%H A257903 Clark Kimberling, <a href="/A257903/b257903.txt">Table of n, a(n) for n = 1..1000</a>
%F A257903 a(k+1) - a(k) = d(k+1) for k >= 1.
%e A257903 a(1) = 0, d(1) = 3;
%e A257903 a(2) = 1, d(2) = 1;
%e A257903 a(3) = 3, d(3) = 2;
%e A257903 a(4) = 2, d(4) = -1.
%t A257903 a[1] = 0; d[1] = 3; k = 1; z = 10000; zz = 120;
%t A257903 A[k_] := Table[a[i], {i, 1, k}]; diff[k_] := Table[d[i], {i, 1, k}];
%t A257903 c[k_] := Complement[Range[-z, z], diff[k]];
%t A257903 T[k_] := -a[k] + Complement[Range[z], A[k]]
%t A257903 Table[{h = Min[Intersection[c[k], T[k]]], a[k + 1] = a[k] + h,
%t A257903    d[k + 1] = h, k = k + 1}, {i, 1, zz}];
%t A257903 u = Table[a[k], {k, 1, zz}]  (* A257903 *)
%t A257903 Table[d[k], {k, 1, zz}]      (* A257904 *)
%Y A257903 Cf. A257904, A257883, A257705.
%K A257903 nonn,easy
%O A257903 1,3
%A A257903 _Clark Kimberling_, May 13 2015