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A304497 Solution (a(n)) of the system of complementary equations defined in Comments. 6
1, 3, 6, 8, 10, 13, 15, 17, 20, 22, 24, 27, 29, 31, 34, 36, 38, 41, 43, 45, 48, 50, 53, 55, 57, 59, 62, 64, 66, 69, 71, 73, 76, 78, 80, 83, 85, 87, 90, 92, 94, 97, 99, 101, 104, 106, 108, 111, 113, 116, 118, 120, 122, 125, 127, 129, 132, 134, 136, 139, 141 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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. A304498, A304499, A304500.

Sequence in context: A169863 A304500 A047282 * A189937 A190325 A064437

Adjacent sequences:  A304494 A304495 A304496 * A304498 A304499 A304500

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, May 16 2018

STATUS

approved

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Last modified November 29 18:41 EST 2021. Contains 349416 sequences. (Running on oeis4.)