OFFSET
0,2
COMMENTS
Define sequences a(n), b(n), c(n) recursively:
a(n) = least new;
b(n) = least new > = a(n) + 2;
c(n) = a(n) + b(n) - 2;
where "least new k" means the least positive integer not yet placed.
***
The sequences a,b,c partition the positive integers.
***
Conjectures: for n >=0,
0 <= 5*n + 4 - 2*a(n) <= 5,
0 <= 5*n + 8 - 2*b(n) <= 4,
0 <= c(n) - 5n <= 4.
LINKS
Clark Kimberling, Table of n, a(n) for n = 0..1000
EXAMPLE
n: 0 1 2 3 4 5 6 7 8 9 10
a: 1 4 5 9 12 13 16 17 21 27 28
b: 3 6 7 11 14 15 19 20 23 25 29
c: 2 8 10 18 24 26 33 35 42 45 54
MATHEMATICA
z = 300;
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = b = c = {};
Do[{AppendTo[a,
mex[Flatten[{a, b, c}], If[Length[a] == 0, 1, Last[a]]]],
AppendTo[b, mex[Flatten[{a, b, c}], Last[a] + 2]],
AppendTo[c, Last[a] + Last[b] - 2]}, {z}];
Take[a, 100] (* A297291 *)
Take[b, 100] (* A297292 *)
Take[c, 100] (* A297293 *)
(* Peter J. C. Moses, Apr 23 2018 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Apr 24 2018
STATUS
approved