A298868 Solution (a(n)) of the system of 3 complementary equations in Comments.

1, 4, 6, 8, 11, 14, 15, 17, 19, 21, 24, 26, 27, 29, 32, 33, 34, 37, 41, 42, 45, 46, 48, 52, 53, 54, 57, 58, 59, 61, 64, 67, 70, 72, 73, 74, 77, 79, 82, 83, 87, 90, 92, 93, 94, 96, 98, 100, 101, 104, 105, 107, 111, 113, 115, 118, 119, 120, 122, 125, 126, 127

Offset

0,2

Comments

Define sequences a(n), b(n), c(n) recursively, starting with a(0) = 1, b(0) = 2:

a(n) = least new;

b(n) = least new k >= a(n) + n;

c(n) = a(n) + b(n);

where "least new k" means the least positive integer not yet placed.

***

The sequences a,b,c partition the positive integers. Let x = be the greatest solution of 1/x + 1/(x+1) + 1/(2x+1) = 1. Then

x = 1/3 + (2/3)*sqrt(7)*cos((1/3)*arctan((3*sqrt(111))/67));

x = 2.07816258732933084676..., and a(n)/n -> x, b(n)/n -> x+1, and c(n)/n -> 2x+1.

Links

Clark Kimberling, Table of n, a(n) for n = 0..1000

Example

n:   0    1    2    3    4    5    6    7    8    9
a:   1    4    6    8   11   14   15   17   19   21
b:   2    5    7   10   12   16   20   22   25   28
c:   3    9   13   18   23   30   35   39   44   49

Mathematica

z = 400;
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = {1}; b = {2}; c = {}; AppendTo[c, Last[a] + Last[b]]; n = 0;
Do[{n++, AppendTo[a, mex[Flatten[{a, b, c}], 1]],
   AppendTo[b, mex[Flatten[{a, b, c}], a[[n]] + n]],
   AppendTo[c, Last[a] + Last[b]]}, {z}];
Take[a, 100] (* A298868 *)
Take[b, 100] (* A298869 *)
Take[c, 100] (* A298870 *)
(* Peter J. C. Moses, Apr 08 2018 *)

Crossrefs

Cf. A299634, A298869, A298870.

Sequence in context: A003662 A132635 A182131 * A134779 A226250 A075325

Adjacent sequences: A298865 A298866 A298867 * A298869 A298870 A298871

Keyword

nonn,easy

Author

Clark Kimberling, Apr 17 2018

Status

approved