A228805 Number of distinct denominators of rational numbers whose continued fraction consists exclusively of 1s and 2s and has length <=n.

1, 2, 4, 8, 14, 26, 45, 85, 155, 286, 540, 1047, 2014, 3939, 7642, 15082, 29601, 58797, 116248, 231530, 459976, 917030, 1826669, 3641525, 7253511, 14463688, 28834305, 57481793, 114543879, 228286471

Offset

1,2

Links

Table of n, a(n) for n=1..30.

Example

a = 1 counts the denominators in {1/1,2/1} corresponding to the continued fractions [1] and [2]; a(2) = 2 counts the denominators in {1/1,2/1,2/1,3/2,3/1,5/2} corresponding to the continued fractions [1], [2], [1,1], [1,2], [2,1], [2,2].

Mathematica

a[n_] := Length[Sort[Union[Denominator[Map[FromContinuedFraction, Flatten[Map[Tuples[{1, 2}, #] &, Range[n]], 1]]]]]]; Table[a[n], {n, 1, 12}]

Prog

(Sage)
import itertools
def a(n):
    ans = set()
    for k in range(1, n+1):
        for p in itertools.product([1, 2], repeat=k):
            ans.add(continued_fraction(p).value().denominator())
    return len(ans)  # Robin Visser, Nov 19 2023

Crossrefs

Cf. A228803.

Sequence in context: A027557 A120545 A130708 * A317884 A365253 A054193

Adjacent sequences: A228802 A228803 A228804 * A228806 A228807 A228808

Keyword

nonn,easy,changed

Author

Clark Kimberling, Sep 04 2013

Extensions

a(23)-a(30) from Robin Visser, Nov 19 2023

Status

approved