This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A360565 #16 Feb 12 2023 10:02:10 %S A360565 2,3,4,3,5,6,5,7,5,8,7,5,9,7,10,9,8,7,11,7,12,11,8,7,13,11,9,4,14,13, %T A360565 11,15,13,11,16,15,14,13,12,11,17,13,11,18,17,14,13,12,11,19,17,13,11, %U A360565 20,19,17,13,11,21,19,17,13,6,22,21,20,19,18,17,14,13,23,19,17,13,24,23,19,18,17,14,13,25,23,10,19,9,17,15 %N A360565 Denominators of breadth-first numerator-denominator-incrementing enumeration of rationals in (0,1). %C A360565 Construct a tree of rational numbers by starting with a root labeled 1/2. Then iteratively add children to each node breadth-first as follows: to the node labeled p/q in lowest terms, add children labeled with any of p/(q+1) and (p+1)/q (in that order) that are less than one and have not already appeared in the tree. Then a(n) is the denominator of the n-th rational number (in lowest terms) added to the tree. %C A360565 This construction is similar to the Farey tree except that the children of p/q are its mediants with 0/1 and 1/0 (if those mediants have not already occurred), rather than its mediants with its nearest neighbors among its ancestors. %C A360565 For a proof that the tree described above includes all rational numbers between 0 and 1, see Gordon and Whitney. %H A360565 Glen Whitney, <a href="/A360565/b360565.txt">Table of n, a(n) for n = 1..10052</a> %H A360565 G. Gordon and G. Whitney, <a href="https://www.jstor.org/stable/48664225">The Playground Problem 367</a>, Math Horizons, Vol. 26 No. 1 (2018), 32-33. %e A360565 To build the tree, 1/2 only has child 1/3, since 2/2 = 1 is outside of (0,1). Then 1/3 has children 1/4 and 2/3. In turn, 1/4 only has child 1/5 because 2/4 = 1/2 has already occurred, and 2/3 has no children because 2/4 has already occurred and 3/3 is too large. Thus, the sequence begins 2, 3, 4, 3, 5, ... (the denominators of 1/2, 1/3, 1/4, 2/3, 1/5, ...). %o A360565 (Python) # See the entry for A360564. %Y A360565 Numerators in A360564. %Y A360565 Level sizes of the tree in A360566. %Y A360565 See also the Farey tree in A007305 and A007306. %Y A360565 Cf. A293248. %K A360565 frac,nonn,tabf,look %O A360565 1,1 %A A360565 _Glen Whitney_, Feb 11 2023