A304498 Solution (b(n)) of the system of complementary equations defined in Comments.

2, 5, 7, 9, 12, 14, 16, 18, 21, 23, 26, 28, 30, 33, 35, 37, 39, 42, 44, 47, 49, 51, 54, 56, 58, 60, 63, 65, 68, 70, 72, 75, 77, 79, 81, 84, 86, 89, 91, 93, 96, 98, 100, 102, 105, 107, 110, 112, 114, 117, 119, 121, 123, 126, 128, 131, 133, 135, 138, 140, 142

Offset

0,1

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. A304497, A304499, A304500.

Sequence in context: A184397 A024671 A246706 * A184743 A184749 A047386

Adjacent sequences: A304495 A304496 A304497 * A304499 A304500 A304501

Keyword

nonn,easy

Author

Clark Kimberling, May 16 2018

Status

approved