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%I #9 Nov 20 2023 08:19:24
%S 1,2,4,8,14,26,45,85,155,286,540,1047,2014,3939,7642,15082,29601,
%T 58797,116248,231530,459976,917030,1826669,3641525,7253511,14463688,
%U 28834305,57481793,114543879,228286471
%N Number of distinct denominators of rational numbers whose continued fraction consists exclusively of 1s and 2s and has length <=n.
%e 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].
%t a[n_] := Length[Sort[Union[Denominator[Map[FromContinuedFraction, Flatten[Map[Tuples[{1, 2}, #] &, Range[n]], 1]]]]]]; Table[a[n], {n, 1, 12}]
%o (Sage)
%o import itertools
%o def a(n):
%o ans = set()
%o for k in range(1, n+1):
%o for p in itertools.product([1,2], repeat=k):
%o ans.add(continued_fraction(p).value().denominator())
%o return len(ans) # _Robin Visser_, Nov 19 2023
%Y Cf. A228803.
%K nonn,easy
%O 1,2
%A _Clark Kimberling_, Sep 04 2013
%E a(23)-a(30) from _Robin Visser_, Nov 19 2023
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