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A299634 Solution (a(n)) of the system of 3 complementary equations in Comments.

%I A299634 #8 May 05 2018 04:18:19
%S A299634 1,4,10,12,14,17,23,26,30,37,40,42,49,50,54,58,62,64,67,68,70,76,78,
%T A299634 86,88,90,92,95,102,104,106,110,112,118,120,126,131,138,142,144,147,
%U A299634 150,158,160,162,164,168,170,174,182,186,192,196,198,201,210,215,218
%N A299634 Solution (a(n)) of the system of 3 complementary equations in Comments.
%C A299634 Define sequences a(n), b(n), c(n) recursively, starting with a(0) = 1, b(0) = 2:
%C A299634 a(n) = least new k >= 2*b(n-1);
%C A299634 b(n) = least new k;
%C A299634 c(n) = a(n) + b(n);
%C A299634 where "least new k" means the least positive integer not yet placed.
%C A299634 ***
%C A299634 The sequences a,b,c partition the positive integers.
%C A299634 ***
%C A299634 Let x = 11/6.  Conjectures:
%C A299634 a(n) - 2*n*x = 0 for infinitely many n;
%C A299634 b(n) - n*x = 0 for infinitely many n;
%C A299634 c(n) - 3*n*x = 0 for infinitely many n;
%C A299634 (a(n) - 2*n*x) is unbounded below and above;
%C A299634 (b(n) - n*x) is unbounded below and above;
%C A299634 (c(n) - 3*n*x) is unbounded below and above;
%C A299634 ***
%C A299634 Let d(a), d(b), d(c) denote the respective difference sequences. Conjectures:
%C A299634 12 occurs infinitely many times in d(a); 6 occurs infinitely many times in d(b);
%C A299634 2 occurs infinitely many times in d(c).
%H A299634 Clark Kimberling, <a href="/A299634/b299634.txt">Table of n, a(n) for n = 0..1000</a>
%e A299634 n:   0    1    2    3    4    5    6    7    8    9
%e A299634 a:   1    4   10   12   14   17   23   26   30   37
%e A299634 b:   2    5    6    7    8   11   13   15   18   20
%e A299634 c:   3    9   16   19   22   28   36   41   48   57
%t A299634 z = 1000;
%t A299634 mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
%t A299634 a = {1}; b = {2}; c = {}; AppendTo[c, Last[a] + Last[b]];
%t A299634 Do[{
%t A299634   AppendTo[a, mex[Flatten[{a, b, c}], 2 Last[b]]],
%t A299634   AppendTo[b, mex[Flatten[{a, b, c}], 1]],
%t A299634   AppendTo[c, Last[a] + Last[b]]}, {z}];
%t A299634 Take[a, 100] (* A299634 *)
%t A299634 Take[b, 100] (* A299635 *)
%t A299634 Take[c, 100] (* A299636 *)
%t A299634 (* _Peter J. C. Moses_, Apr 08 2018 *)
%Y A299634 Cf. A299635, A299636.
%K A299634 nonn,easy
%O A299634 0,2
%A A299634 _Clark Kimberling_, Apr 17 2018