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

%I #8 May 05 2018 04:18:46

%S 3,9,13,18,23,30,35,39,44,49,55,62,65,69,75,80,84,88,97,102,108,112,

%T 116,123,129,132,138,143,145,150,155,162,169,175,179,183,187,193,199,

%U 204,211,218,225,228,231,235,240,246,249,255,259,263,270,277,282,288

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

%C Define sequences a(n), b(n), c(n) recursively, starting with a(0) = 1, b(0) = 2:

%C a(n) = least new;

%C b(n) = least new k >= a(n) + n;

%C c(n) = a(n) + b(n);

%C where "least new k" means the least positive integer not yet placed.

%C ***

%C 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

%C x = 1/3 + (2/3)*sqrt(7)*cos((1/3)*arctan((3*sqrt(111))/67))

%C x = 2.07816258732933084676..., and a(n)/n - > x, b(n)/n -> x+1, and c(n)/n - > 2x+1.

%H Clark Kimberling, <a href="/A298870/b298870.txt">Table of n, a(n) for n = 0..1000</a>

%e n: 0 1 2 3 4 5 6 7 8 9

%e a: 1 4 6 8 11 14 15 17 19 21

%e b: 2 5 7 10 12 16 20 22 25 28

%e c: 3 9 13 18 23 30 35 39 44 49

%t z = 400;

%t mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);

%t a = {1}; b = {2}; c = {}; AppendTo[c, Last[a] + Last[b]]; n = 0;

%t Do[{n++, AppendTo[a, mex[Flatten[{a, b, c}], 1]],

%t AppendTo[b, mex[Flatten[{a, b, c}], a[[n]] + n]],

%t AppendTo[c, Last[a] + Last[b]]}, {z}];

%t Take[a, 100] (* A298868 *)

%t Take[b, 100] (* A298869 *)

%t Take[c, 100] (* A298870 *)

%t (* _Peter J. C. Moses_, Apr 08 2018 *)

%Y Cf. A299634, A298868, A298869.

%K nonn,easy

%O 0,1

%A _Clark Kimberling_, Apr 18 2018