cp's OEIS Frontend

A299636 Solution (c(n)) of the system of 3 complementary equations in Comments.

%I A299636 #8 May 05 2018 04:18:27
%S A299636 3,9,16,19,22,28,36,41,48,57,61,66,74,77,83,89,94,97,101,103,108,115,
%T A299636 121,130,133,136,139,146,154,157,161,166,171,178,183,191,200,209,214,
%U A299636 217,222,229,238,241,244,248,253,257,265,275,282,290,295,298,306,317
%N A299636 Solution (c(n)) of the system of 3 complementary equations in Comments.
%C A299636 Define sequences a(n), b(n), c(n) recursively, starting with a(0) = 1, b(0) = 2:
%C A299636 a(n) = least new k >= 2*b(n-1);
%C A299636 b(n) = least new k;
%C A299636 c(n) = a(n) + b(n);
%C A299636 where "least new k" means the least positive integer not yet placed.
%C A299636 ***
%C A299636 The sequences a,b,c partition the positive integers.
%C A299636 ***
%C A299636 Let x = 11/6.  Conjectures:
%C A299636 a(n) - 2*n*x = 0 for infinitely many n;
%C A299636 b(n) - n*x = 0 for infinitely many n;
%C A299636 c(n) - 3*n*x = 0 for infinitely many n;
%C A299636 (a(n) - 2*n*x) is unbounded below and above;
%C A299636 (b(n) - n*x) is unbounded below and above;
%C A299636 (c(n) - 3*n*x) is unbounded below and above;
%C A299636 ***
%C A299636 Let d(a), d(b), d(c) denote the respective difference sequences. Conjectures:
%C A299636 12 occurs infinitely many times in d(a); 6 occurs infinitely many times in d(b);
%C A299636 2 occurs infinitely many times in d(c).
%H A299636 Clark Kimberling, <a href="/A299636/b299636.txt">Table of n, a(n) for n = 0..1000</a>
%e A299636 n:   0    1    2    3    4    5    6    7    8    9
%e A299636 a:   1    4   10   12   14   17   23   26   30   37
%e A299636 b:   2    5    6    7    8   11   13   15   18   20
%e A299636 c:   3    9   16   19   22   28   36   41   48   57
%t A299636 z = 1000;
%t A299636 mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
%t A299636 a = {1}; b = {2}; c = {}; AppendTo[c, Last[a] + Last[b]];
%t A299636 Do[{
%t A299636   AppendTo[a, mex[Flatten[{a, b, c}], 2 Last[b]]],
%t A299636   AppendTo[b, mex[Flatten[{a, b, c}], 1]],
%t A299636   AppendTo[c, Last[a] + Last[b]]}, {z}];
%t A299636 Take[a, 100] (* A299634 *)
%t A299636 Take[b, 100] (* A299635 *)
%t A299636 Take[c, 100] (* A299636 *)
%t A299636 (* _Peter J. C. Moses_, Apr 08 2018 *)
%Y A299636 Cf. A299634, A299635.
%K A299636 nonn,easy
%O A299636 0,1
%A A299636 _Clark Kimberling_, Apr 17 2018