A335999 a(1) = 1; for n >= 2, a(n) = least positive integer not in {a(1),..., a(n-1), b(1),...,b(n-1)}, where for n >=1, b(n) = n + 2 + least positive integer not in {a(1),..., a(n-1), a(n), b(1),...,b(n-1)}.

1, 2, 3, 4, 6, 8, 10, 11, 13, 14, 16, 17, 19, 21, 22, 24, 26, 27, 29, 31, 32, 34, 35, 37, 39, 40, 42, 43, 45, 47, 48, 50, 51, 53, 55, 56, 58, 60, 61, 63, 64, 66, 68, 69, 71, 73, 74, 76, 77, 79, 81, 82, 84, 86, 87, 89, 90, 92, 94, 95, 97, 98, 100, 102, 103

Offset

1,2

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.

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Guide to related sequences:

k sequences (a(n)) and (b(n))

0 A000201 and A001950 (lower and upper Wythoff sequences)

1 A026351 and A026352

2 A022342 and A003622

3 A335999 and A336008

Links

Table of n, a(n) for n=1..65.

Example

a(1) = 1; b(1) = 1+2+2 = 5
a(2) = 2; b(2) = 2+2+3 = 7
a(3) = 3; b(3) = 3+2+4 = 9
a(4) = 4; b(4) = 4+2+6 = 12

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

Cf. A336008.

Sequence in context: A257457 A122138 A047418 * A026508 A184103 A352318

Adjacent sequences: A335996 A335997 A335998 * A336000 A336001 A336002

Keyword

nonn

Author

Clark Kimberling, Jul 16 2020

Status

approved