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A304502 Solution (c(n)) of the system of complementary equations defined in Comments. 3
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 (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) = 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: A033913 A088124 A029456 * A190743 A331133 A110208

Adjacent sequences:  A304499 A304500 A304501 * A304503 A304504 A304505

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, May 19 2018

STATUS

approved

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Last modified May 30 15:21 EDT 2020. Contains 334726 sequences. (Running on oeis4.)