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A297291 Solution (a(n)) of the system of 3 complementary equations in Comments. 3
1, 4, 5, 9, 12, 13, 16, 17, 21, 22, 27, 28, 31, 32, 37, 38, 41, 44, 47, 48, 51, 52, 57, 58, 61, 62, 67, 68, 71, 72, 77, 78, 81, 84, 85, 89, 90, 93, 97, 98, 101, 104, 107, 108, 111, 112, 117, 118, 121, 122, 127, 128, 131, 132, 137, 138, 141, 144, 147, 148 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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.

LINKS

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

EXAMPLE

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

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 *)

CROSSREFS

Cf. A299634, A297292, A297293.

Sequence in context: A319606 A230239 A194154 * A269741 A047610 A126004

Adjacent sequences:  A297288 A297289 A297290 * A297292 A297293 A297294

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Apr 24 2018

STATUS

approved

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Last modified September 20 03:47 EDT 2021. Contains 347577 sequences. (Running on oeis4.)