A304497 Solution (a(n)) of the system of complementary equations defined in Comments.

1, 3, 6, 8, 10, 13, 15, 17, 20, 22, 24, 27, 29, 31, 34, 36, 38, 41, 43, 45, 48, 50, 53, 55, 57, 59, 62, 64, 66, 69, 71, 73, 76, 78, 80, 83, 85, 87, 90, 92, 94, 97, 99, 101, 104, 106, 108, 111, 113, 116, 118, 120, 122, 125, 127, 129, 132, 134, 136, 139, 141

Offset

0,2

Comments

Define sequences a(n), b(n), c(n) recursively, starting with a(0) = 1:

a(n) = least new,

b(n) = least new,

c(n) = 2*a(n) + b(n),

where "least new k" means the least positive integer not yet placed. The three sequences partition the positive integers. Empirically, for all n >= 0,

1 <= 3*a(n) - 7*n <= 5,

5 <= 3*b(n) - 7*n <= 8,

3 <= c(n) - 7*n <= 6.

Links

Table of n, a(n) for n=0..60.

Example

a(0) = 1, b(0) = 2; c(0) = 2*1 + 2 = 4, so that a(1) = 3, so that b(1) = 4, so that c(1) = 11.

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]]];
  AppendTo[c, 2 Last[a] + Last[b]], {z}];
Take[a, 100] (* A304497 *)
Take[b, 100] (* A304498 *)
Take[c, 100] (* A304499 *)
Grid[{Join[{"n"}, Range[0, 20]], Join[{"a(n)"}, Take[a, 21]],
  Join[{"b(n)"}, Take[b, 21]], Join[{"c(n)"}, Take[c, 21]]},
Alignment -> ".",  Dividers -> {{2 -> Red, -1 -> Blue}, {2 -> Red, -1 -> Blue}}]
(* Peter J. C. Moses, Apr 26 2018 *)

Crossrefs

Cf. A304498, A304499, A304500.

Sequence in context: A169863 A304500 A047282 * A189937 A190325 A064437

Adjacent sequences: A304494 A304495 A304496 * A304498 A304499 A304500

Keyword

nonn,easy

Author

Clark Kimberling, May 16 2018

Status

approved