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

%I A194738 #5 Mar 30 2012 18:57:43
%S A194738 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,
%T A194738 29,21,13,5,30,21,12,3,31,21,11,1,32,21,10,43,31,19,7,43,30,17,4,43,
%U A194738 29,15,56,41,26,11,55,39,23,7,54,37,20,3,53,35,17,69,50,31,12,67
%N A194738 Number of k such that {k*sqrt(3)} < {n*sqrt(3)}, where { } = fractional part.
%C A194738 Related sequences:
%C A194738 A019587, A194733, A019588, A194734; |r|=(1+sqrt(5))/2
%C A194738 A054072, A194735, A194736, A194737; |r|=sqrt(2)
%C A194738 A194738, A194739, A194740, A194741; |r|=sqrt(3)
%C A194738 A194742, A194743, A194744, A194745; |r|=sqrt(5)
%C A194738 A194746, A194747, A194748, A194749; |r|=sqrt(6)
%C A194738 A194750, A194751, A194752, A194753; |r|=e
%C A194738 A194754, A194755, A194756, A194757; |r|=pi
%C A194738 A194758, A194759, A194760, A194761; |r|=log(2)
%C A194738 A194762, A194763, A194764, A194765; |r|=2^(1/3)
%C A194738 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.
%e A194738 {r}=0.7...; {2r}=0.4...; {3r}=0.1...;
%e A194738 {4f}=0.9...; {5r}=0.6...; so that a(5)=3.
%t A194738 r = Sqrt[3]; p[x_] := FractionalPart[x];
%t A194738 u[n_, k_] := If[p[k*r] <= p[n*r], 1, 0]
%t A194738 v[n_, k_] := If[p[k*r] > p[n*r], 1, 0]
%t A194738 s[n_] := Sum[u[n, k], {k, 1, n}]
%t A194738 t[n_] := Sum[v[n, k], {k, 1, n}]
%t A194738 Table[s[n], {n, 1, 100}]   (* A194738 *)
%t A194738 Table[t[n], {n, 1, 100}]   (* A194739 *)
%Y A194738 Cf. A194739, A194740.
%K A194738 nonn
%O A194738 1,4
%A A194738 _Clark Kimberling_, Sep 02 2011