A336008 Complement of A335999.

5, 7, 9, 12, 15, 18, 20, 23, 25, 28, 30, 33, 36, 38, 41, 44, 46, 49, 52, 54, 57, 59, 62, 65, 67, 70, 72, 75, 78, 80, 83, 85, 88, 91, 93, 96, 99, 101, 104, 106, 109, 112, 114, 117, 120, 122, 125, 127, 130, 133, 135, 138, 141, 143, 146, 148, 151, 154, 156, 159

Offset: 1

Author

Clark Kimberling, Jul 16 2020

Keywords

• nonn

Comments

In general, let u(1) = 1, and let k be a positive integer. Define u(n) = least positive integer not in {u(1), ..., u(n-1), v(1), ..., v(n-1)} and v(n) = n - 1 + k + least positive integer not in {u(1), ..., u(n-1), u(n), v(1), ..., v(n-1)}. As sets, (u(n)) and (v(n)) are disjoint. If k >= -1, let a(n) = u(n) and b(n) = v(n) for all n >= 1, but if k <= -2, let a(n) = u(n) - k + 1 and b(n) = v(n) - k - 1 for all n >= 1. Then every positive integer is in exactly one of the sequences (a(n)) and (b(n)). The difference sequence of (a(n)) consists of 1's and 2's; the difference sequence of (b(n)) consists of 2's and 3's. See A335999 for a guide to related sequences.

Crossrefs

Cf. A335999.

Programs

Mathematica

mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
{a, b} = {{1}, {}}; k = 3;
Do[AppendTo[b, Length[b] + k + mex[Flatten[{a, b}], Last[a]]];
AppendTo[a, mex[Flatten[{a, b}], Last[a]]], {150}]
a  (* A335999 *)
b  (* A336008 *)
(* Peter J. C. Moses, Jul 13 2020 *)

Extensions

Corrected by Clark Kimberling, Sep 26 2020