A298870 Solution (c(n)) of the system of 3 complementary equations in Comments.

3, 9, 13, 18, 23, 30, 35, 39, 44, 49, 55, 62, 65, 69, 75, 80, 84, 88, 97, 102, 108, 112, 116, 123, 129, 132, 138, 143, 145, 150, 155, 162, 169, 175, 179, 183, 187, 193, 199, 204, 211, 218, 225, 228, 231, 235, 240, 246, 249, 255, 259, 263, 270, 277, 282, 288

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, A298869.

Sequence in context: A163595 A088090 A075326 * A095234 A240240 A032415

Adjacent sequences: A298867 A298868 A298869 * A298871 A298872 A298873

Keyword

nonn,easy

Author

Clark Kimberling, Apr 18 2018

Status

approved