A299635 Solution (b(n)) of the system of 3 complementary equations in Comments.

2, 5, 6, 7, 8, 11, 13, 15, 18, 20, 21, 24, 25, 27, 29, 31, 32, 33, 34, 35, 38, 39, 43, 44, 45, 46, 47, 51, 52, 53, 55, 56, 59, 60, 63, 65, 69, 71, 72, 73, 75, 79, 80, 81, 82, 84, 85, 87, 91, 93, 96, 98, 99, 100, 105, 107, 109, 111, 113, 114, 116, 117, 119

Offset

0,1

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. A299634, A299636.

Sequence in context: A122546 A258775 A275894 * A170944 A074940 A028752

Adjacent sequences: A299632 A299633 A299634 * A299636 A299637 A299638

Keyword

nonn,easy

Author

Clark Kimberling, Apr 17 2018

Status

approved