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A297292
Solution (b(n)) of the system of 3 complementary equations in Comments.
3
3, 6, 7, 11, 14, 15, 19, 20, 23, 25, 29, 30, 34, 36, 39, 40, 43, 46, 49, 50, 53, 55, 59, 60, 64, 65, 69, 70, 73, 75, 79, 80, 83, 86, 87, 91, 92, 95, 99, 100, 103, 106, 109, 110, 113, 115, 119, 120, 124, 126, 129, 130, 133, 135, 139, 140, 143, 146, 149, 150
OFFSET
0,1
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
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