A298869 Solution (b(n)) of the system of 3 complementary equations in Comments.

2, 5, 7, 10, 12, 16, 20, 22, 25, 28, 31, 36, 38, 40, 43, 47, 50, 51, 56, 60, 63, 66, 68, 71, 76, 78, 81, 85, 86, 89, 91, 95, 99, 103, 106, 109, 110, 114, 117, 121, 124, 128, 133, 135, 137, 139, 142, 146, 148, 151, 154, 156, 159, 164, 167, 170, 174, 176, 178

Offset

0,1

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, A298868, A298870.

Sequence in context: A206807 A182773 A346155 * A219666 A026340 A331833

Adjacent sequences: A298866 A298867 A298868 * A298870 A298871 A298872

Keyword

nonn,easy

Author

Clark Kimberling, Apr 17 2018

Status

approved