A298418 Solution (c(n)) of the system of 3 complementary equations in Comments.

3, 10, 13, 18, 23, 31, 37, 40, 43, 50, 53, 63, 68, 70, 73, 82, 85, 89, 91, 98, 111, 115, 118, 120, 129, 139, 141, 144, 150, 153, 155, 160, 164, 168, 183, 187, 189, 194, 198, 201, 203, 217, 219, 232, 235, 240, 247, 253, 255, 258, 261, 264, 268, 270, 275, 284

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 > = a(n) + n + 1;

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.

(The same limits occur in A298868 and A297469.)

Links

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

Example

n:   0   1   2   3   4    5   6   7   8   9  10
a:   1   4   5   7   9   12  15  16  17  20  21
b:   2   6   8  11   14  19  22  24  26  30  32
c:   3  10  13  18   23  31  37  40  43  50  53

Mathematica

z=200;
mex[list_, start_]:=(NestWhile[#+1&, start, MemberQ[list, #]&]);
a={1}; b={2}; c={3}; n=0;
Do[{n++;
  AppendTo[a, mex[Flatten[{a, b, c}], If[Length[a]==0, 1, Last[a]]]],
  AppendTo[b, mex[Flatten[{a, b, c}], Last[a]+n+1]],
  AppendTo[c, Last[a]+Last[b]]}, {z}];
Take[a, 100] (* A297838 *)
Take[b, 100] (* A298170 *)
Take[c, 100] (* A298418 *)
(* Peter J. C. Moses, Apr 23 2018 *)

Crossrefs

Cf. A299634, A298868, A297469, A298838, A298418.

Sequence in context: A299983 A188373 A190776 * A229205 A229483 A235089

Adjacent sequences: A298415 A298416 A298417 * A298419 A298420 A298421

Keyword

nonn,easy

Author

Clark Kimberling, May 01 2018

Status

approved