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

Author

Clark Kimberling, Sep 03 2011

Keywords

• nonn

Comments

A194813 + A194814 = A000027 for n > 0.

Examples

			{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.
		

Crossrefs

Cf. A001622, A194814, A194738.

Partial sums of A327174.

Programs

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 *)