A298170 Solution (b(n)) of the system of 3 complementary equations in Comments.

2, 6, 8, 11, 14, 19, 22, 24, 26, 30, 32, 38, 41, 42, 44, 49, 51, 54, 55, 59, 66, 69, 71, 72, 77, 83, 84, 86, 90, 92, 93, 96, 99, 101, 109, 112, 113, 116, 119, 121, 122, 130, 131, 138, 140, 143, 147, 151, 152, 154, 156, 158, 161, 162, 165, 170, 174, 181, 184

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, A297838, A298418.

Sequence in context: A298474 A285971 A079418 * A067246 A370866 A184420

Adjacent sequences: A298167 A298168 A298169 * A298171 A298172 A298173

Keyword

nonn,easy

Author

Clark Kimberling, Apr 25 2018

Status

approved