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A257918 Sequence (d(n)) generated by Rule 1 (in Comments) with a(1) = 2 and d(1) = 2.

%I A257918 #6 May 14 2015 12:00:42
%S A257918 2,-1,3,1,-2,4,5,-3,6,-4,-5,7,8,-7,-6,9,10,-8,-9,12,11,-10,13,-11,14,
%T A257918 -13,15,-12,16,-14,-15,18,-16,17,19,-17,20,-19,-18,22,21,-20,23,-22,
%U A257918 24,-21,-23,25,26,-24,27,-25,28,-26,-27,30,-28,29,31,-29,32,-31
%N A257918 Sequence (d(n)) generated by Rule 1 (in Comments) with a(1) = 2 and d(1) = 2.
%C A257918 This is the sequence (d(n)) of differences associated with the sequence a = A257882.
%C A257918 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 A257918 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 A257918 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 A257918 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 A257918 See A257705 for a guide to related sequences.
%H A257918 Clark Kimberling, <a href="/A257918/b257918.txt">Table of n, a(n) for n = 1..1000</a>
%F A257918 d(k) = a(k) - a(k-1) for k >=2, where a(k) = A257882(k).
%e A257918 a(1) = 2, d(1) = 2;
%e A257918 a(2) = 1, d(2) = -1;
%e A257918 a(3) = 4, d(3) = 3;
%e A257918 a(4) = 5, d(4) = 1.
%t A257918 a[1] = 2; d[1] = 2; k = 1; z = 10000; zz = 120;
%t A257918 A[k_] := Table[a[i], {i, 1, k}]; diff[k_] := Table[d[i], {i, 1, k}];
%t A257918 c[k_] := Complement[Range[-z, z], diff[k]];
%t A257918 T[k_] := -a[k] + Complement[Range[z], A[k]];
%t A257918 s[k_] := Intersection[Range[-a[k], -1], c[k], T[k]];
%t A257918 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 A257918 u = Table[a[k], {k, 1, zz}]  (* A257882 *)
%t A257918 Table[d[k], {k, 1, zz}]      (* A257918 *)
%Y A257918 Cf. A131389, A257705, A081145, A257918, A175499.
%K A257918 easy,sign
%O A257918 1,1
%A A257918 _Clark Kimberling_, May 13 2015