A304451 Solution (a(n)) of the complementary equation in Comments.

2, 4, 8, 9, 12, 15, 17, 19, 21, 23, 28, 30, 33, 35, 38, 39, 41, 43, 47, 48, 52, 54, 57, 60, 64, 66, 69, 70, 74, 75, 77, 79, 82, 83, 87, 88, 92, 93, 95, 97, 101, 102, 107, 108, 110, 113, 118, 120, 123, 126, 129, 130, 134, 135, 137, 140, 145, 147, 149, 150

Offset

0,1

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..59.

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. A304452.

Sequence in context: A297126 A044814 A342009 * A098374 A182852 A318083

Adjacent sequences: A304448 A304449 A304450 * A304452 A304453 A304454

Keyword

nonn,easy

Author

Clark Kimberling, May 16 2018

Status

approved