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

0,1

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)

Links

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

Example

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

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 *)

Crossrefs

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

Sequence in context: A063205 A090613 A063097 * A356190 A168421 A038942

Adjacent sequences: A297466 A297467 A297468 * A297470 A297471 A297472

Keyword

nonn,easy

Author

Clark Kimberling, May 04 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

Status

approved