login
A298870
Solution (c(n)) of the system of 3 complementary equations in Comments.
3
3, 9, 13, 18, 23, 30, 35, 39, 44, 49, 55, 62, 65, 69, 75, 80, 84, 88, 97, 102, 108, 112, 116, 123, 129, 132, 138, 143, 145, 150, 155, 162, 169, 175, 179, 183, 187, 193, 199, 204, 211, 218, 225, 228, 231, 235, 240, 246, 249, 255, 259, 263, 270, 277, 282, 288
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;
b(n) = least new k >= a(n) + n;
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 = be the greatest solution of 1/x + 1/(x+1) + 1/(2x+1) = 1. Then
x = 1/3 + (2/3)*sqrt(7)*cos((1/3)*arctan((3*sqrt(111))/67))
x = 2.07816258732933084676..., and a(n)/n - > x, b(n)/n -> x+1, and c(n)/n - > 2x+1.
LINKS
EXAMPLE
n: 0 1 2 3 4 5 6 7 8 9
a: 1 4 6 8 11 14 15 17 19 21
b: 2 5 7 10 12 16 20 22 25 28
c: 3 9 13 18 23 30 35 39 44 49
MATHEMATICA
z = 400;
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = {1}; b = {2}; c = {}; AppendTo[c, Last[a] + Last[b]]; n = 0;
Do[{n++, AppendTo[a, mex[Flatten[{a, b, c}], 1]],
AppendTo[b, mex[Flatten[{a, b, c}], a[[n]] + n]],
AppendTo[c, Last[a] + Last[b]]}, {z}];
Take[a, 100] (* A298868 *)
Take[b, 100] (* A298869 *)
Take[c, 100] (* A298870 *)
(* Peter J. C. Moses, Apr 08 2018 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Apr 18 2018
STATUS
approved