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

1,4

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.

Links

Table of n, a(n) for n=1..75.

Example

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

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 *)

Crossrefs

Cf. A194739, A194740.

Sequence in context: A085064 A030587 A194764 * A194750 A370350 A194743

Adjacent sequences: A194735 A194736 A194737 * A194739 A194740 A194741

Keyword

nonn

Author

Clark Kimberling, Sep 02 2011

Status

approved