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A194738
Number of k such that {k*sqrt(3)} < {n*sqrt(3)}, where { } = fractional part.
34
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)
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.
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
Sequence in context: A085064 A030587 A194764 * A194750 A370350 A194743
KEYWORD
nonn
AUTHOR
Clark Kimberling, Sep 02 2011
STATUS
approved