A194735 Number of positive integers k <= n such that {k*sqrt(2)} > {n*sqrt(2)}, where { } = fractional part.

0, 0, 2, 1, 4, 2, 0, 5, 2, 8, 4, 0, 8, 3, 12, 6, 16, 9, 2, 14, 6, 19, 10, 1, 16, 6, 22, 11, 28, 16, 4, 23, 10, 30, 16, 2, 24, 9, 32, 16, 0, 25, 8, 34, 16, 43, 24, 5, 34, 14, 44, 23, 2, 34, 12, 45, 22, 56, 32, 8, 44, 19, 56, 30, 4, 43, 16, 56, 28, 0, 42, 13, 56, 26, 70, 39

Offset: 1

Author

Clark Kimberling, Sep 02 2011

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A054072, A194738.

Programs

Mathematica

r = Sqrt[2]; 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}]   (* A054072 *)
Table[t[n], {n, 1, 100}]   (* A194735 *)