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A298869
Solution (b(n)) of the system of 3 complementary equations in Comments.
3
2, 5, 7, 10, 12, 16, 20, 22, 25, 28, 31, 36, 38, 40, 43, 47, 50, 51, 56, 60, 63, 66, 68, 71, 76, 78, 81, 85, 86, 89, 91, 95, 99, 103, 106, 109, 110, 114, 117, 121, 124, 128, 133, 135, 137, 139, 142, 146, 148, 151, 154, 156, 159, 164, 167, 170, 174, 176, 178
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 17 2018
STATUS
approved