A304502 Solution (c(n)) of the system of complementary equations defined in Comments.

5, 11, 20, 26, 34, 41, 47, 53, 61, 68, 74, 83, 89, 95, 103, 110, 116, 124, 131, 137, 146, 152, 160, 167, 173, 179, 188, 194, 200, 209, 215, 223, 230, 236, 242, 250, 257, 263, 272, 278, 286, 293, 299, 305, 314, 320, 326, 335, 341, 349, 356, 362, 368, 377, 383

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) = a(n) + 2*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 <= 4,

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

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

Links

Table of n, a(n) for n=0..54.

Example

a(0) = 1, b(0) = 2; c(0) = 1 + 2*2 = 5, 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, Last[a] + 2*Last[b]], {z}];
Take[a, 100] (* A304500 *)
Take[b, 100] (* A304501 *)
Take[c, 100] (* A304502 *)
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, A304500, A304501.

Sequence in context: A344549 A345353 A029456 * A190743 A331133 A110208

Adjacent sequences: A304499 A304500 A304501 * A304503 A304504 A304505

Keyword

nonn,easy

Author

Clark Kimberling, May 19 2018

Status

approved