OFFSET

1,1

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.

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 *)

CROSSREFS

KEYWORD

nonn

AUTHOR

Clark Kimberling, Jul 16 2020

EXTENSIONS

Corrected by Clark Kimberling, Sep 26 2020

STATUS

approved