A299636 Solution (c(n)) of the system of 3 complementary equations in Comments.

3, 9, 16, 19, 22, 28, 36, 41, 48, 57, 61, 66, 74, 77, 83, 89, 94, 97, 101, 103, 108, 115, 121, 130, 133, 136, 139, 146, 154, 157, 161, 166, 171, 178, 183, 191, 200, 209, 214, 217, 222, 229, 238, 241, 244, 248, 253, 257, 265, 275, 282, 290, 295, 298, 306, 317

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, A299635.

Sequence in context: A336970 A366051 A070066 * A271491 A359618 A214644

Adjacent sequences: A299633 A299634 A299635 * A299637 A299638 A299639

Keyword

nonn,easy

Author

Clark Kimberling, Apr 17 2018

Status

approved