A297293 Solution (c(n)) of the system of 3 complementary equations in Comments.

2, 8, 10, 18, 24, 26, 33, 35, 42, 45, 54, 56, 63, 66, 74, 76, 82, 88, 94, 96, 102, 105, 114, 116, 123, 125, 134, 136, 142, 145, 154, 156, 162, 168, 170, 178, 180, 186, 194, 196, 202, 208, 214, 216, 222, 225, 234, 236, 243, 246, 254, 256, 262, 265, 274, 276

Offset

0,1

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, A297291, A297292.

Sequence in context: A197115 A288824 A190042 * A294157 A082396 A195582

Adjacent sequences: A297290 A297291 A297292 * A297294 A297295 A297296

Keyword

nonn,easy

Author

Clark Kimberling, Apr 24 2018

Status

approved