A299634 Solution (a(n)) of the system of 3 complementary equations in Comments.

1, 4, 10, 12, 14, 17, 23, 26, 30, 37, 40, 42, 49, 50, 54, 58, 62, 64, 67, 68, 70, 76, 78, 86, 88, 90, 92, 95, 102, 104, 106, 110, 112, 118, 120, 126, 131, 138, 142, 144, 147, 150, 158, 160, 162, 164, 168, 170, 174, 182, 186, 192, 196, 198, 201, 210, 215, 218

Offset

0,2

Comments

Define sequences a(n), b(n), c(n) recursively, starting with a(0) = 1, b(0) = 2:

a(n) = least new k >= 2*b(n-1);

b(n) = least new k;

c(n) = a(n) + b(n);

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

***

The sequences a,b,c partition the positive integers.

***

Let x = 11/6. Conjectures:

a(n) - 2*n*x = 0 for infinitely many n;

b(n) - n*x = 0 for infinitely many n;

c(n) - 3*n*x = 0 for infinitely many n;

(a(n) - 2*n*x) is unbounded below and above;

(b(n) - n*x) is unbounded below and above;

(c(n) - 3*n*x) is unbounded below and above;

***

Let d(a), d(b), d(c) denote the respective difference sequences. Conjectures:

12 occurs infinitely many times in d(a); 6 occurs infinitely many times in d(b);

2 occurs infinitely many times in d(c).

Links

Clark Kimberling, Table of n, a(n) for n = 0..1000

Example

n:   0    1    2    3    4    5    6    7    8    9
a:   1    4   10   12   14   17   23   26   30   37
b:   2    5    6    7    8   11   13   15   18   20
c:   3    9   16   19   22   28   36   41   48   57

Mathematica

z = 1000;
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = {1}; b = {2}; c = {}; AppendTo[c, Last[a] + Last[b]];
Do[{
  AppendTo[a, mex[Flatten[{a, b, c}], 2 Last[b]]],
  AppendTo[b, mex[Flatten[{a, b, c}], 1]],
  AppendTo[c, Last[a] + Last[b]]}, {z}];
Take[a, 100] (* A299634 *)
Take[b, 100] (* A299635 *)
Take[c, 100] (* A299636 *)
(* Peter J. C. Moses, Apr 08 2018 *)

Crossrefs

Cf. A299635, A299636.

Sequence in context: A071179 A155475 A023693 * A181053 A239055 A295129

Adjacent sequences: A299631 A299632 A299633 * A299635 A299636 A299637

Keyword

nonn,easy

Author

Clark Kimberling, Apr 17 2018

Status

approved