A299423 -id:A299423 - cp's OEIS frontend

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

Clark Kimberling, May 01 2018

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.)

			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

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

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

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 *)

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

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