A194734 Number of k such that {-k*r} > {-n*r}, where { } = fractional part, r = (1+sqrt(5))/2 (the golden ratio).

0, 0, 2, 1, 0, 4, 2, 7, 4, 1, 8, 4, 0, 9, 4, 14, 8, 2, 14, 7, 20, 12, 4, 19, 10, 1, 18, 8, 26, 15, 4, 24, 12, 0, 22, 9, 32, 18, 4, 29, 14, 40, 24, 8, 36, 19, 2, 32, 14, 45, 26, 7, 40, 20, 54, 33, 12, 48, 26, 4, 42, 19, 58, 34, 10, 51, 26, 1, 44, 18, 62, 35, 8, 54, 26, 73, 44

Offset: 1

Author

Clark Kimberling, Sep 02 2011

Keywords

• nonn

Comments

The fractional part here uses the Mma implementation for negative arguments: It is the fractional part of the absolute value, turned negative. So a(n) = A019587(n)-1. - R. J. Mathar, Aug 13 2021

Crossrefs

Cf. A019588, A194738.

Programs

Mathematica

r = -GoldenRatio; 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}]   (* A019588 *)
Table[t[n], {n, 1, 100}]   (* A193734 *)