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A189405
n+[ns/r]+[nt/r]; r=1, s=-1+sqrt(3), t=1+sqrt(3).
4
3, 8, 13, 16, 21, 26, 31, 34, 39, 44, 49, 52, 57, 62, 65, 70, 75, 80, 83, 88, 93, 98, 101, 106, 111, 116, 119, 124, 129, 132, 137, 142, 147, 150, 155, 160, 165, 168, 173, 178, 183, 186, 191, 196, 199, 204, 209, 214, 217, 222, 227, 232, 235, 240, 245, 248, 253, 258, 263, 266, 271, 276, 281, 284, 289, 294, 299, 302, 307, 312, 315, 320, 325, 330, 333, 338, 343, 348, 351, 356, 361, 366, 369
OFFSET
1,1
COMMENTS
This is one of three sequences that partition the positive integers. In general, suppose that r, s, t are positive real numbers for which the sets {i/r: i>=1}, {j/s: j>=1}, {k/t: k>=1} are pairwise disjoint. Let a(n) be the rank of n/r when all the numbers in the three sets are jointly ranked. Define b(n) and c(n) as the ranks of n/s and n/t. It is easy to prove that
a(n)=n+[ns/r]+[nt/r],
b(n)=n+[nr/s]+[nt/s],
c(n)=n+[nr/t]+[ns/t], where []=floor.
Taking s=-1+sqrt(3), t=1+sqrt(3) gives
MATHEMATICA
r=1; s=-1+3^(1/2); t=1+3^(1/2);
a[n_] := n + Floor[n*s/r] + Floor[n*t/r];
b[n_] := n + Floor[n*r/s] + Floor[n*t/s];
c[n_] := n + Floor[n*r/t] + Floor[n*s/t];
Table[a[n], {n, 1, 120}] (*A189405*)
Table[b[n], {n, 1, 120}] (*A189406*)
Table[c[n], {n, 1, 120}] (*A189407*)
CROSSREFS
Sequence in context: A310297 A310298 A189519 * A355125 A310299 A310300
KEYWORD
nonn
AUTHOR
Clark Kimberling, Apr 21 2011
STATUS
approved