A297291 Solution (a(n)) of the system of 3 complementary equations in Comments.

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

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