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A299634
Solution (a(n)) of the system of 3 complementary equations in Comments.
35
1, 4, 10, 12, 14, 17, 23, 26, 30, 37, 40, 42, 49, 50, 54, 58, 62, 64, 67, 68, 70, 76, 78, 86, 88, 90, 92, 95, 102, 104, 106, 110, 112, 118, 120, 126, 131, 138, 142, 144, 147, 150, 158, 160, 162, 164, 168, 170, 174, 182, 186, 192, 196, 198, 201, 210, 215, 218
OFFSET
0,2
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(n-1);
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}];
Take[a, 100] (* A299634 *)
Take[b, 100] (* A299635 *)
Take[c, 100] (* A299636 *)
(* Peter J. C. Moses, Apr 08 2018 *)
CROSSREFS
Sequence in context: A071179 A155475 A023693 * A181053 A239055 A295129
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Apr 17 2018
STATUS
approved