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A304499 Solution (c(n)) of the system of complementary equations defined in Comments. 3
4, 11, 19, 25, 32, 40, 46, 52, 61, 67, 74, 82, 88, 95, 103, 109, 115, 124, 130, 137, 145, 151, 160, 166, 172, 178, 187, 193, 200, 208, 214, 221, 229, 235, 241, 250, 256, 263, 271, 277, 284, 292, 298, 304, 313, 319, 326, 334, 340, 349, 355, 361, 367, 376, 382 (list; graph; refs; listen; history; text; internal format)
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:

   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..54.

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, A304498, A304500.

Sequence in context: A301108 A339214 A008061 * A063215 A278709 A247988

Adjacent sequences:  A304496 A304497 A304498 * A304500 A304501 A304502

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, May 16 2018

STATUS

approved

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Last modified January 24 21:49 EST 2022. Contains 350565 sequences. (Running on oeis4.)