A178047 -id:A178047 - cp's OEIS frontend

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

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, 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, 10, 10, 10, 10, 10, 10, 10, 10, 10

Offset: 1

N. J. A. Sloane, Dec 16 2010

nonn

Computed by Alan Wechsler, Dec 16 2010.
Richard C. Schroeppel also asked about the analogous sequence giving the last occurrence of denominator n.
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

			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:
1:   0/1                                     1/1
2:   0/1                 1/2                 1/1
3:   0/1       1/3       1/2       2/3       1/1
4:   0/1  1/4  1/3  2/5  1/2  3/5  2/3  3/4  1/1
We only look at the denominators in this table (which form the sequence A049456, or A002487 if the rightmost column is removed).
1 first appears in row 1, so a(1) = 1.
2 first appears in row 2, so a(2) = 2.
3 first appears in row 3, so a(3) = 3.
4 and 5 first appear in row 4, so a(4) = a(5) = 4.

Based on a posting by Richard C. Schroeppel to the Math Fun Mailing List, Dec 15 2010.

Bo Gyu Jeong, Table of n, a(n) for n = 1..10000
Richard J. Mathar, The Kepler binary tree of reduced fractions, 2017.

See A178047 for another version. Cf. A002487, A006842, A006843, A177903, A178042, A135510.

More terms from Bo Gyu Jeong, Oct 20 2012

A295783 First differences of A293160.

Original entry on oeis.org

0, 1, 1, 2, 2, 6, 7, 11, 17, 30, 40, 73, 109, 165, 269, 441, 675, 1076, 1671, 2699, 4256, 6726, 10585, 16969, 26524, 42386, 66979

Offset: 1

R. J. Mathar, Nov 27 2017

a(n) is the number of distinct numerators that exist in row n of the Kepler tree A294442 but not yet in row n-1 of the tree (assuming a row count such that 1/1 is in row 0).
It is the number of numerators that are "new" in row n (because the set of denominators of row n-1 contributes to the set of numerators of row n).
a(n) is nonnegative because A293160 is monotonically increasing (because all numerators of one row become numerators of the next row).
Define the "entry level" E(j) as the smallest row number at which denominator j appears in A294442 (again: row counts start at 1/1 as row 0), then a(n+1) is the number of occurrences of n in j: a(n+1) = #{j: E(j)=n}.
E(j) = A178047(j), as originally observed by R. J. Mathar, because every denominator j first appears both in Kepler's tree (used in E(j)) and in the left half of Stern-Brocot tree (used in A178047) when there is a fraction p/q with p+q=j in the previous row, and the rows of these two trees contain the same fractions (in different orders), assuming the row labeling from A178047 for Stern-Brocot tree. - Andrey Zabolotskiy, Dec 06 2024

R. J. Mathar, The Kepler tree of reduced fractions, (2017).

Cf. A294442, A294443, A178047.

Differences@ Map[Length@ Union@ Numerator@ # &, #] &@ Nest[Append[#, Flatten@ Map[{#1/(#1 + #2), #2/(#1 + #2)} & @@ {Numerator@ #, Denominator@ #} &, Last@ #]] &, {{1/1}, {1/2}}, 21] (* Michael De Vlieger, Apr 18 2018 *)

a(25)-a(27) from Michael De Vlieger, Apr 18 2018

cp's OEIS Frontend

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

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