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A297838 Solution (a(n)) of the system of 3 complementary equations in Comments. 5
1, 4, 5, 7, 9, 12, 15, 16, 17, 20, 21, 25, 27, 28, 29, 33, 34, 35, 36, 39, 45, 46, 47, 48, 52, 56, 57, 58, 60, 61, 62, 64, 65, 67, 74, 75, 76, 78, 79, 80, 81, 87, 88, 94, 95, 97, 100, 102, 103, 104, 105, 106, 107, 108, 110, 114, 117, 123, 124, 125, 126, 127 (list; graph; refs; listen; history; text; internal format)
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;
b(n) = least new > = a(n) + n + 1;
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.
(The same limits occur in A298868 and A297469.)
LINKS
EXAMPLE
n: 0 1 2 3 4 5 6 7 8 9 10
a: 1 4 5 7 9 12 15 16 17 20 21
b: 2 6 8 11 14 19 22 24 26 30 32
c: 3 10 13 18 23 31 37 40 43 50 53
MATHEMATICA
z=200;
mex[list_, start_]:=(NestWhile[#+1&, start, MemberQ[list, #]&]);
a={1}; b={2}; c={3}; n=0;
Do[{n++;
AppendTo[a, mex[Flatten[{a, b, c}], If[Length[a]==0, 1, Last[a]]]],
AppendTo[b, mex[Flatten[{a, b, c}], Last[a]+n+1]],
AppendTo[c, Last[a]+Last[b]]}, {z}];
Take[a, 100] (* A297838 *)
Take[b, 100] (* A298170 *)
Take[c, 100] (* A298418 *)
(* Peter J. C. Moses, Apr 23 2018 *)
CROSSREFS
Sequence in context: A286050 A047493 A285307 * A032360 A117150 A239276
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Apr 25 2018
STATUS
approved

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Last modified July 22 00:52 EDT 2024. Contains 374478 sequences. (Running on oeis4.)