A299636 - OEIS

%I #8 May 05 2018 04:18:27

%S 3,9,16,19,22,28,36,41,48,57,61,66,74,77,83,89,94,97,101,103,108,115,

%T 121,130,133,136,139,146,154,157,161,166,171,178,183,191,200,209,214,

%U 217,222,229,238,241,244,248,253,257,265,275,282,290,295,298,306,317

%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 k >= 2*b(n-1);

%C b(n) = least new k;

%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.

%C ***

%C Let x = 11/6.  Conjectures:

%C a(n) - 2*n*x = 0 for infinitely many n;

%C b(n) - n*x = 0 for infinitely many n;

%C c(n) - 3*n*x = 0 for infinitely many n;

%C (a(n) - 2*n*x) is unbounded below and above;

%C (b(n) - n*x) is unbounded below and above;

%C (c(n) - 3*n*x) is unbounded below and above;

%C ***

%C Let d(a), d(b), d(c) denote the respective difference sequences. Conjectures:

%C 12 occurs infinitely many times in d(a); 6 occurs infinitely many times in d(b);

%C 2 occurs infinitely many times in d(c).

%H Clark Kimberling, <a href="/A299636/b299636.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   10   12   14   17   23   26   30   37

%e b:   2    5    6    7    8   11   13   15   18   20

%e c:   3    9   16   19   22   28   36   41   48   57

%t z = 1000;

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

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

%t Do[{

%t   AppendTo[a, mex[Flatten[{a, b, c}], 2 Last[b]]],

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

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

%t Take[a, 100] (* A299634 *)

%t Take[b, 100] (* A299635 *)

%t Take[c, 100] (* A299636 *)

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

%Y Cf. A299634, A299635.

%K nonn,easy

%O 0,1

%A _Clark Kimberling_, Apr 17 2018