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A297838 Solution (a(n)) of the system of 3 complementary equations in Comments. 5
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified January 21 18:52 EST 2021. Contains 340352 sequences. (Running on oeis4.)