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A304498 Solution (b(n)) of the system of complementary equations defined in Comments. 3
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 (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, 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

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Last modified August 5 01:51 EDT 2021. Contains 346456 sequences. (Running on oeis4.)