A297469 Solution (bb(n)) of the system of 3 complementary equations in Comments.

2, 7, 11, 17, 22, 27, 31, 37, 41, 47, 51, 57, 62, 67, 71, 77, 82, 87, 91, 97, 102, 107, 111, 117, 121, 127, 131, 137, 142, 147, 151, 157, 161, 167, 171, 177, 182, 187, 191, 197, 201, 207, 211, 217, 222, 227, 231, 237, 242, 247, 251, 257, 262, 267, 271, 277

Offset: 0

Author

Clark Kimberling, May 04 2018

Keywords

• nonn
• easy

Comments

Define sequences aa(n), bb(n), cc(n) recursively, starting with aa(0) = 1, bb(0) = 2, cc(0) = 3:

aa(n) = least new;

bb(n) = aa(n) + cc(n-1);

cc(n) = least new;

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

***

The sequences aa,bb,cc partition the positive integers. It appears that cc = A047218 and that for every n >= 0,

(1) 5*n - 1 - 2*aa(n) is in {0,1,2},

(2) (aa(n) mod 5) is in {1,2,4},

(3) 5*n - 3 - bb(n) is in {0,1} for every n >= 0;

(4) (bb(n) mod 5) is in {1,2}.

From N. J. A. Sloane, Nov 05 2019: (Start)

Conjecture: For t >= 0, bb(2t) = 10t + 1 (+1 if binary expansion of t ends in an odd number of 0's), bb(2t+1) = 10t + 7.

The first part may also be written as bb(2t) = 10t + 1 + A328789(t-1).

(End)

Examples

			n:  0 1 2 3 4 5 6 7 8 9 10
aa: 1 4 6 9 12 14 16 19 21 24 26
bb: 2 7 11 17 22 27 31 37 41 47 51
cc: 3 5 8 10 13 15 18 20 23 25 28
		

Links

• Clark Kimberling, Table of n, a(n) for n = 0..10000 [This is the sequence bb]

Crossrefs

Cf. A299634, A298468 (aa), A047218 (cc), A328789.

Programs

Mathematica

z = 500;
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = {1}; b = {2}; c = {3};
Do[AppendTo[a, mex[Flatten[{a, b, c}], Last[a]]];
  AppendTo[b, Last[a] + Last[c]];
  AppendTo[c, mex[Flatten[{a, b, c}], Last[a]]], {z}];
Take[a, 100] (* A298468 *)
Take[b, 100] (* A297469 *)
Take[c, 100] (* A047218 *)
(* Peter J. C. Moses, Apr 23 2018 *)

Extensions

Changed a,b,c to aa,bb,cc to avoid confusion caused by conflict with standard OEIS terminology. - N. J. A. Sloane, Nov 03 2019