A299533 Solution (b(n)) of the system of 3 complementary equations in Comments.

3, 5, 10, 13, 15, 18, 20, 23, 27, 30, 34, 38, 40, 43, 47, 50, 53, 55, 59, 62, 64, 68, 70, 73, 77, 79, 82, 86, 89, 92, 93, 99, 101, 104, 108, 111, 114, 115, 120, 122, 126, 129, 132, 135, 140, 142, 144, 146, 150, 153, 157, 160, 163, 165, 169, 174, 176, 178

Offset

0,1

Comments

Define sequences a(n), b(n), c(n) recursively:

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 A297838.)

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   2   6   8   9  11  12  14  17  19  22
b:   3   5  10  13  15  18  20  23  27  30  34
c:   4   7  16  21  24  29  32  37  44  49  56

Mathematica

z = 200;
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = {}; b = {}; c = {}; 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}];
(* Peter J. C. Moses, Apr 23 2018 *)
Take[a, 100] (* A297469 *)
Take[b, 100] (* A299533 *)
Take[c, 100] (* A299423 *)
(* Peter J. C. Moses, Apr 23 2018 *)

Crossrefs

Cf. A299634, A298868, A297838, A297469, A299423.

Sequence in context: A345176 A275219 A080931 * A309270 A165718 A340528

Adjacent sequences: A299530 A299531 A299532 * A299534 A299535 A299536

Keyword

nonn,easy

Author

Clark Kimberling, May 01 2018

Status

approved