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A257985 Sequence (a(n)) generated by Rule 3 (in Comments) with a(1) = 2 and d(1) = 0.

%I A257985 #4 Jun 05 2015 03:42:49
%S A257985 2,3,5,4,7,13,6,11,8,15,9,17,12,10,19,37,14,27,16,31,18,35,20,39,21,
%T A257985 41,22,43,23,45,24,47,25,49,32,28,55,26,51,42,30,59,29,57,33,65,34,67,
%U A257985 40,44,36,71,38,75,50,99,46,91,48,95,53,105,54,107,52,103
%N A257985 Sequence (a(n)) generated by Rule 3 (in Comments) with a(1) = 2 and d(1) = 0.
%C A257985 Rule 3 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 A257985 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 least such h, let a(k+1) = a(k) + h, replace k by k + 1, and repeat Step 1; otherwise do Step 2.
%C A257985 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 A257985 See A257905 for a guide to related sequences and conjectures.
%H A257985 Clark Kimberling, <a href="/A257985/b257985.txt">Table of n, a(n) for n = 1..1000</a>
%F A257985 a(n) = A131393(n) + 1 for n >= 1. Also, a(n) - a(n-1) = A131394(n) for n >= 2.
%e A257985 a(1) = 2, d(1) = 0;
%e A257985 a(2) = 3, d(2) = 1;
%e A257985 a(3) = 5, d(3) = 2;
%e A257985 a(4) = 4, d(4) = -1.
%t A257985 {a, f} = {{2}, {0}}; Do[tmp = {#, # - Last[a]} &[Min[Complement[#, Intersection[a, #]]&[Last[a] + Complement[#, Intersection[f, #]] &[Range[2 - Last[a], -1]]]]];
%t A257985 If[! IntegerQ[tmp[[1]]], tmp = {Last[a] + #, #} &[NestWhile[# + 1 &, 1, ! (! MemberQ[f, #] && ! MemberQ[a, Last[a] - #]) &]]]; AppendTo[a, tmp[[1]]]; AppendTo[f, tmp[[2]]], {120}]; {a, f} (* _Peter J. C. Moses_, May 14 2015 *)
%Y A257985 Cf. A257905, A257986, A131394.
%K A257985 nonn,easy
%O A257985 1,1
%A A257985 _Clark Kimberling_, Jun 02 2015