A304452 Solution (b(n)) of the complementary equation in Comments.

1, 3, 5, 6, 7, 10, 11, 13, 14, 16, 18, 20, 22, 24, 25, 26, 27, 29, 31, 32, 34, 36, 37, 40, 42, 44, 45, 46, 49, 50, 51, 53, 55, 56, 58, 59, 61, 62, 63, 65, 67, 68, 71, 72, 73, 76, 78, 80, 81, 84, 85, 86, 89, 90, 91, 94, 96, 98, 99, 100, 103, 104, 105, 106

Offset

0,2

Comments

Define sequences a(n) and b(n) recursively, starting with b(0) = 1:

b(n) = least new;

a(n) = b(n) + b(floor(n/2)),

where "least new k" means the least positive integer not yet placed.

***

Conjectures: a(n)/n -> 5/2 and -1 <= 5/2 - a(n)/n <= 2 for n >= 1;

b(n)/n -> 5/3 and -1 <= 5/3 - b(n)/n <= 2 for n >= 1.

Links

Table of n, a(n) for n=0..63.

Example

a(0) = b(0) + b(0) = 2;
a(1) = b(1) + b(2) >= 3 + 4, so that b(2) = 3, b(2) = 4, b(3) = 5, b(4) = 6, and a(1) = 7.

Mathematica

mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
z = 1000; a = {}; b = {1};
Do[AppendTo[a, Last[b] + b[[Floor[(1 + Length[b])/2]]]];
  AppendTo[b, mex[Flatten[{a, b}], 1]], {z}];
Take[a, 100]  (* A304451 *)
Take[b, 100]  (* A304452 *)
(* Peter J. C. Moses, May 10 2018 *)

Crossrefs

Cf. A304451.

Sequence in context: A336353 A182851 A281995 * A292763 A176175 A157201

Adjacent sequences: A304449 A304450 A304451 * A304453 A304454 A304455

Keyword

nonn,easy

Author

Clark Kimberling, May 16 2018

Status

approved