A194813 Number of integers k in [1,n] such that {n*r + k*r} < {n*r - k*r}, where { } = fractional part and r = (1+sqrt(5))/2 (the golden ratio).

0, 0, 1, 2, 2, 2, 3, 4, 5, 5, 5, 6, 6, 6, 6, 7, 8, 8, 8, 9, 10, 11, 11, 12, 13, 13, 13, 13, 14, 15, 15, 15, 16, 16, 16, 16, 17, 18, 18, 18, 19, 20, 21, 21, 22, 23, 23, 23, 23, 24, 25, 25, 25, 26, 27, 28, 28, 29, 30, 30, 30, 31, 32, 33, 33, 33, 34, 34, 34, 34, 35, 36, 36

Offset

1,4

Comments

A194813 + A194814 = A000027 for n > 0.

Links

Table of n, a(n) for n=1..73.

Example

{4r+1r} = 0.09...; {4r-1r} = 0.85...;
{4r+2r} = 0.70...; {4r-2r} = 0.23...;
{4r+3r} = 0.32...; {4r-3r} = 0.61...;
{4r+4r} = 0.94...; {4r-4r} = 0.00...;
so that a(4)=2.

Mathematica

r = GoldenRatio; p[x_] := FractionalPart[x];
u[n_, k_] := If[p[n*r + k*r] <= p[n*r - k*r], 1, 0]
v[n_, k_] := If[p[n*r + k*r] > p[n*r - k*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}]   (* A194813 *)
Table[t[n], {n, 1, 100}]   (* A194814 *)

Crossrefs

Cf. A001622, A194814, A194738.

Partial sums of A327174.

Sequence in context: A330918 A162352 A285509 * A096533 A230413 A029096

Adjacent sequences: A194810 A194811 A194812 * A194814 A194815 A194816

Keyword

nonn

Author

Clark Kimberling, Sep 03 2011

Status

approved