A299423 Solution (c(n)) of the system of 3 complementary equations in Comments.

4, 7, 16, 21, 24, 29, 32, 37, 44, 49, 56, 63, 66, 71, 78, 83, 88, 91, 98, 103, 106, 113, 116, 121, 128, 131, 136, 143, 147, 152, 154, 164, 168, 173, 180, 185, 189, 191, 200, 203, 210, 214, 219, 225, 234, 237, 240, 243, 250, 255, 262, 267, 272, 275, 281, 291

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, A299533.

Sequence in context: A256926 A101534 A110933 * A067398 A325657 A054599

Adjacent sequences: A299420 A299421 A299422 * A299424 A299425 A299426

Keyword

nonn,easy

Author

Clark Kimberling, May 01 2018

Status

approved