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A298468 Solution (aa(n)) of the system of 3 complementary equations in Comments. 7
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 (list; graph; refs; listen; history; text; internal format)
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: A258743 A292660 A190081 * A190304 A189366 A066095

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

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Last modified May 8 08:51 EDT 2021. Contains 343666 sequences. (Running on oeis4.)