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A299635 Solution (b(n)) of the system of 3 complementary equations in Comments.

%I A299635 #8 May 05 2018 04:18:23
%S A299635 2,5,6,7,8,11,13,15,18,20,21,24,25,27,29,31,32,33,34,35,38,39,43,44,
%T A299635 45,46,47,51,52,53,55,56,59,60,63,65,69,71,72,73,75,79,80,81,82,84,85,
%U A299635 87,91,93,96,98,99,100,105,107,109,111,113,114,116,117,119
%N A299635 Solution (b(n)) of the system of 3 complementary equations in Comments.
%C A299635 Define sequences a(n), b(n), c(n) recursively, starting with a(0) = 1, b(0) = 2:
%C A299635 a(n) = least new k >= 2*b(n-1);
%C A299635 b(n) = least new k;
%C A299635 c(n) = a(n) + b(n);
%C A299635 where "least new k" means the least positive integer not yet placed.
%C A299635 ***
%C A299635 The sequences a,b,c partition the positive integers.
%C A299635 ***
%C A299635 Let x = 11/6.  Conjectures:
%C A299635 a(n) - 2*n*x = 0 for infinitely many n;
%C A299635 b(n) - n*x = 0 for infinitely many n;
%C A299635 c(n) - 3*n*x = 0 for infinitely many n;
%C A299635 (a(n) - 2*n*x) is unbounded below and above;
%C A299635 (b(n) - n*x) is unbounded below and above;
%C A299635 (c(n) - 3*n*x) is unbounded below and above;
%C A299635 ***
%C A299635 Let d(a), d(b), d(c) denote the respective difference sequences. Conjectures:
%C A299635 12 occurs infinitely many times in d(a); 6 occurs infinitely many times in d(b);
%C A299635 2 occurs infinitely many times in d(c).
%H A299635 Clark Kimberling, <a href="/A299635/b299635.txt">Table of n, a(n) for n = 0..1000</a>
%e A299635 n:   0    1    2    3    4    5    6    7    8    9
%e A299635 a:   1    4   10   12   14   17   23   26   30   37
%e A299635 b:   2    5    6    7    8   11   13   15   18   20
%e A299635 c:   3    9   16   19   22   28   36   41   48   57
%t A299635 z = 1000;
%t A299635 mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
%t A299635 a = {1}; b = {2}; c = {}; AppendTo[c, Last[a] + Last[b]];
%t A299635 Do[{
%t A299635   AppendTo[a, mex[Flatten[{a, b, c}], 2 Last[b]]],
%t A299635   AppendTo[b, mex[Flatten[{a, b, c}], 1]],
%t A299635   AppendTo[c, Last[a] + Last[b]]}, {z}];
%t A299635 Take[a, 100] (* A299634 *)
%t A299635 Take[b, 100] (* A299635 *)
%t A299635 Take[c, 100] (* A299636 *)
%t A299635 (* _Peter J. C. Moses_, Apr 08 2018 *)
%Y A299635 Cf. A299634, A299636.
%K A299635 nonn,easy
%O A299635 0,1
%A A299635 _Clark Kimberling_, Apr 17 2018