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A178031 Consider the Farey tree A049455/A049456; a(n) = row at which the denominator n first appears (assumes first row is labeled row 1).

%I A178031 #38 Jun 29 2025 21:39:06
%S A178031 1,2,3,4,4,6,5,5,6,6,6,6,6,7,7,7,7,7,7,8,7,8,8,8,8,8,8,9,8,8,8,9,9,8,
%T A178031 9,9,9,10,9,9,9,10,9,9,9,9,9,10,9,9,10,10,10,11,9,10,10,10,10,10,10,
%U A178031 10,10,10,10,10,10,10,10,10
%N A178031 Consider the Farey tree A049455/A049456; a(n) = row at which the denominator n first appears (assumes first row is labeled row 1).
%C A178031 Computed by Alan Wechsler, Dec 16 2010.
%C A178031 Richard C. Schroeppel also asked about the analogous sequence giving the last occurrence of denominator n.
%C A178031 The first occurrence of k in this sequence is apparently at n = A135510(k-1), except for k=5. The last occurrence of k is at n = Fibonacci(k). - _Andrey Zabolotskiy_, Dec 01 2024
%D A178031 Based on a posting by Richard C. Schroeppel to the Math Fun Mailing List, Dec 15 2010.
%H A178031 Bo Gyu Jeong, <a href="/A178031/b178031.txt">Table of n, a(n) for n = 1..10000</a>
%H A178031 Richard J. Mathar, <a href="/A294443/a294443.pdf">The Kepler binary tree of reduced fractions</a>, 2017.
%e A178031 Start with a pair of fractions 0/1, 1/1 and repeatedly insert the "Farey sum" (p+r)/(q+s) in between every pair of adjacent fractions p/q, r/s. The first few iterations are:
%e A178031 1:   0/1                                     1/1
%e A178031 2:   0/1                 1/2                 1/1
%e A178031 3:   0/1       1/3       1/2       2/3       1/1
%e A178031 4:   0/1  1/4  1/3  2/5  1/2  3/5  2/3  3/4  1/1
%e A178031 We only look at the denominators in this table (which form the sequence A049456, or A002487 if the rightmost column is removed).
%e A178031 1 first appears in row 1, so a(1) = 1.
%e A178031 2 first appears in row 2, so a(2) = 2.
%e A178031 3 first appears in row 3, so a(3) = 3.
%e A178031 4 and 5 first appear in row 4, so a(4) = a(5) = 4.
%Y A178031 See A178047 for another version. Cf. A002487, A006842, A006843, A177903, A178042, A135510.
%K A178031 nonn
%O A178031 1,2
%A A178031 _N. J. A. Sloane_, Dec 16 2010
%E A178031 More terms from _Bo Gyu Jeong_, Oct 20 2012