A207525 Number of rational numbers p/q such that 0<p<q<=n and p/q<=(the quotient of consecutive Fibonacci numbers which has the greatest denominator <=n).

1, 3, 4, 6, 7, 11, 14, 18, 20, 26, 29, 36, 39, 44, 49, 59, 63, 74, 79, 87, 93, 107, 112, 124, 131, 142, 150, 167, 172, 191, 201, 214, 223, 237, 244, 266, 277, 292, 302, 327, 335, 361, 374, 388, 401, 430, 440, 466, 478, 498, 513, 545, 556, 581, 596, 619

Offset

2,2

Comments

Counts certain consecutive Farey fractions of order n; not the same as A207525, which counts p/q<=(greatest quotient of consecutive Fibonacci numbers having denominator <= n).

Links

Table of n, a(n) for n=2..57.

Example

a(4)=4 counts 1/4, 1/3, 1/2, 2/3.
a(5)=6 counts 1/5, 1/4, 1/3, 2/5, 1/2, 3/5.

Mathematica

r[n_] := Union[Flatten[Table[p/q, {q, 2, n - 1},
  {p, 1, q - 1}]]];
f[n_] := Fibonacci[n]; gr = GoldenRatio;
g[n_] := Floor[Log[gr, 1 + n*Sqrt[5]]];
s = Table[Round[f[g[n - 1]]/gr]/f[g[n - 1]], {n, 3, 82}];
c[n_, x_] := Length[Select[r[n + 2], # <= x &]]
Table[c[n, s[[n]]], {n, 1, 80}]

Crossrefs

Cf. A207524.

Sequence in context: A306678 A075434 A085253 * A374846 A240208 A349555

Adjacent sequences: A207522 A207523 A207524 * A207526 A207527 A207528

Keyword

nonn

Author

Clark Kimberling, Feb 18 2012

Status

approved