A194738 Number of k such that {k*sqrt(3)} < {n*sqrt(3)}, where { } = fractional part.

1, 1, 1, 4, 3, 2, 1, 7, 5, 3, 1, 10, 7, 4, 15, 11, 7, 3, 17, 12, 7, 2, 19, 13, 7, 1, 21, 14, 7, 29, 21, 13, 5, 30, 21, 12, 3, 31, 21, 11, 1, 32, 21, 10, 43, 31, 19, 7, 43, 30, 17, 4, 43, 29, 15, 56, 41, 26, 11, 55, 39, 23, 7, 54, 37, 20, 3, 53, 35, 17, 69, 50, 31, 12, 67

Offset: 1

Author

Clark Kimberling, Sep 02 2011

Keywords

• nonn

Comments

Related sequences:

A019587, A194733, A019588, A194734; |r|=(1+sqrt(5))/2

A054072, A194735, A194736, A194737; |r|=sqrt(2)

A194738, A194739, A194740, A194741; |r|=sqrt(3)

A194742, A194743, A194744, A194745; |r|=sqrt(5)

A194746, A194747, A194748, A194749; |r|=sqrt(6)

A194750, A194751, A194752, A194753; |r|=e

A194754, A194755, A194756, A194757; |r|=pi

A194758, A194759, A194760, A194761; |r|=log(2)

A194762, A194763, A194764, A194765; |r|=2^(1/3)

In each case, trivially, the sum of the first two sequences is A000027(for n>0), and likewise for the sum of the other two.

Examples

			{r}=0.7...; {2r}=0.4...; {3r}=0.1...;
{4f}=0.9...; {5r}=0.6...; so that a(5)=3.
		

Crossrefs

Cf. A194739, A194740.

Programs

Mathematica

r = Sqrt[3]; p[x_] := FractionalPart[x];
u[n_, k_] := If[p[k*r] <= p[n*r], 1, 0]
v[n_, k_] := If[p[k*r] > p[n*r], 1, 0]
s[n_] := Sum[u[n, k], {k, 1, n}]
t[n_] := Sum[v[n, k], {k, 1, n}]
Table[s[n], {n, 1, 100}]   (* A194738 *)
Table[t[n], {n, 1, 100}]   (* A194739 *)