A297838 Solution (a(n)) of the system of 3 complementary equations in Comments.

1, 4, 5, 7, 9, 12, 15, 16, 17, 20, 21, 25, 27, 28, 29, 33, 34, 35, 36, 39, 45, 46, 47, 48, 52, 56, 57, 58, 60, 61, 62, 64, 65, 67, 74, 75, 76, 78, 79, 80, 81, 87, 88, 94, 95, 97, 100, 102, 103, 104, 105, 106, 107, 108, 110, 114, 117, 123, 124, 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 > = 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, A298170, A298418.

Sequence in context: A286050 A047493 A285307 * A032360 A117150 A239276

Adjacent sequences: A297835 A297836 A297837 * A297839 A297840 A297841

Keyword

nonn,easy

Author

Clark Kimberling, Apr 25 2018

Status

approved