A297292 Solution (b(n)) of the system of 3 complementary equations in Comments.

3, 6, 7, 11, 14, 15, 19, 20, 23, 25, 29, 30, 34, 36, 39, 40, 43, 46, 49, 50, 53, 55, 59, 60, 64, 65, 69, 70, 73, 75, 79, 80, 83, 86, 87, 91, 92, 95, 99, 100, 103, 106, 109, 110, 113, 115, 119, 120, 124, 126, 129, 130, 133, 135, 139, 140, 143, 146, 149, 150

Offset: 0

Author

Clark Kimberling, Apr 24 2018

Keywords

• nonn
• easy

Comments

Define sequences a(n), b(n), c(n) recursively:

a(n) = least new;

b(n) = least new > = a(n) + 2;

c(n) = a(n) + b(n) - 2;

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

***

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

***

Conjectures: for n >= 0,

0 <= 5*n + 4 - 2*a(n) <= 5,

0 <= 5*n + 8 - 2*b(n) <= 4,

0 <= c(n) - 5n <= 4.

Examples

			n:   0   1   2   3   4   5   6   7   8   9  10
a:   1   4   5   9  12  13  16  17  21  27  28
b:   3   6   7  11  14  15  19  20  23  25  29
c:   2   8  10  18  24  26  33  35  42  45  54
		

Links

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

Crossrefs

Cf. A299634, A297291, A297293.

Programs

Mathematica

z = 300;
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = b = c = {};
Do[{AppendTo[a,
    mex[Flatten[{a, b, c}], If[Length[a] == 0, 1, Last[a]]]],
   AppendTo[b, mex[Flatten[{a, b, c}], Last[a] + 2]],
   AppendTo[c, Last[a] + Last[b] - 2]}, {z}];
Take[a, 100]  (* A297291 *)
Take[b, 100]  (* A297292 *)
Take[c, 100]  (* A297293 *)
(* Peter J. C. Moses, Apr 23 2018 *)