A298468 Solution (aa(n)) of the system of 3 complementary equations in Comments.

1, 4, 6, 9, 12, 14, 16, 19, 21, 24, 26, 29, 32, 34, 36, 39, 42, 44, 46, 49, 52, 54, 56, 59, 61, 64, 66, 69, 72, 74, 76, 79, 81, 84, 86, 89, 92, 94, 96, 99, 101, 104, 106, 109, 112, 114, 116, 119, 122, 124, 126, 129, 132, 134, 136, 139, 141, 144, 146, 149

Offset

0,2

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 >= 1, aa(2t) = 5t+1(+1 if binary expansion of t ends in an odd number of 0's), and for t >= 0, aa(2t+1) = 5t+4.

The first part may also be written as aa(2t) = 5t+1+A328789(t-1).

(End)

Links

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

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, A297469 (bb), A047218 (cc), A328789.

Sequence in context: A292660 A190081 A364445 * A190304 A367420 A189366

Adjacent sequences: A298465 A298466 A298467 * A298469 A298470 A298471

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