

A299636


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


3



3, 9, 16, 19, 22, 28, 36, 41, 48, 57, 61, 66, 74, 77, 83, 89, 94, 97, 101, 103, 108, 115, 121, 130, 133, 136, 139, 146, 154, 157, 161, 166, 171, 178, 183, 191, 200, 209, 214, 217, 222, 229, 238, 241, 244, 248, 253, 257, 265, 275, 282, 290, 295, 298, 306, 317
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OFFSET

0,1


COMMENTS

Define sequences a(n), b(n), c(n) recursively, starting with a(0) = 1, b(0) = 2:
a(n) = least new k >= 2*b(n1);
b(n) = least new k;
c(n) = a(n) + b(n);
where "least new k" means the least positive integer not yet placed.
***
The sequences a,b,c partition the positive integers.
***
Let x = 11/6. Conjectures:
a(n)  2*n*x = 0 for infinitely many n;
b(n)  n*x = 0 for infinitely many n;
c(n)  3*n*x = 0 for infinitely many n;
(a(n)  2*n*x) is unbounded below and above;
(b(n)  n*x) is unbounded below and above;
(c(n)  3*n*x) is unbounded below and above;
***
Let d(a), d(b), d(c) denote the respective difference sequences. Conjectures:
12 occurs infinitely many times in d(a); 6 occurs infinitely many times in d(b);
2 occurs infinitely many times in d(c).


LINKS



EXAMPLE

n: 0 1 2 3 4 5 6 7 8 9
a: 1 4 10 12 14 17 23 26 30 37
b: 2 5 6 7 8 11 13 15 18 20
c: 3 9 16 19 22 28 36 41 48 57


MATHEMATICA

z = 1000;
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = {1}; b = {2}; c = {}; AppendTo[c, Last[a] + Last[b]];
Do[{
AppendTo[a, mex[Flatten[{a, b, c}], 2 Last[b]]],
AppendTo[b, mex[Flatten[{a, b, c}], 1]],
AppendTo[c, Last[a] + Last[b]]}, {z}];


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



