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

1,3

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

Links

Table of n, a(n) for n=1..77.

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 *)

Crossrefs

Cf. A019588, A194738.

Sequence in context: A002349 A096794 A106375 * A255528 A201701 A131667

Adjacent sequences: A194731 A194732 A194733 * A194735 A194736 A194737

Keyword

nonn

Author

Clark Kimberling, Sep 02 2011

Status

approved