A189750 n+[ns/r]+[nt/r]; r=1, s=arctan(1/3), t=arctan(2/3).

1, 3, 4, 7, 8, 10, 13, 14, 16, 18, 20, 22, 24, 26, 27, 30, 31, 33, 36, 37, 39, 41, 43, 45, 47, 49, 50, 53, 55, 56, 58, 60, 62, 63, 66, 68, 69, 72, 73, 75, 78, 79, 81, 83, 85, 87, 89, 91, 92, 95, 96, 98, 101, 102, 104, 106, 108, 110, 111, 114, 115, 117, 120, 121, 123, 125, 127, 128, 131, 133, 134, 137, 138, 140, 143, 144, 146, 148, 150, 152, 154, 156, 157, 160

Offset

1,2

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 r=1, s=arctan(1/3), t=arctan(2/3) gives

a=A189750, b=A189751, c=A189752.

Links

Table of n, a(n) for n=1..84.

Mathematica

r=1; s=ArcTan[1/3]; t=ArcTan[2/3];
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}]  (*A189750*)
Table[b[n], {n, 1, 120}]  (*A189751*)
Table[c[n], {n, 1, 120}]  (*A189752*)

Crossrefs

Cf. A189751, A189752.

Sequence in context: A285346 A244736 A138971 * A153144 A032788 A338733

Adjacent sequences: A189747 A189748 A189749 * A189751 A189752 A189753

Keyword

nonn

Author

Clark Kimberling, Apr 26 2011

Status

approved