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

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, 41, 22, 43, 23, 45, 24, 47, 25, 49, 32, 28, 55, 26, 51, 42, 30, 59, 29, 57, 33, 65, 34, 67, 40, 44, 36, 71, 38, 75, 50, 99, 46, 91, 48, 95, 53, 105, 54, 107, 52, 103

Offset: 1

Author

Clark Kimberling, Jun 02 2015

Keywords

• nonn
• easy

Comments

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).

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.

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.

See A257905 for a guide to related sequences and conjectures.

Examples

			a(1) = 2, d(1) = 0;
a(2) = 3, d(2) = 1;
a(3) = 5, d(3) = 2;
a(4) = 4, d(4) = -1.
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A257905, A257986, A131394.

Programs

Mathematica

{a, f} = {{2}, {0}}; Do[tmp = {#, # - Last[a]} &[Min[Complement[#, Intersection[a, #]]&[Last[a] + Complement[#, Intersection[f, #]] &[Range[2 - Last[a], -1]]]]];
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 *)

Formula

a(n) = A131393(n) + 1 for n >= 1. Also, a(n) - a(n-1) = A131394(n) for n >= 2.