A360564 Numerators of breadth-first numerator-denominator-incrementing enumeration of rationals in (0,1).

1, 1, 1, 2, 1, 1, 2, 1, 3, 1, 2, 4, 1, 3, 1, 2, 3, 4, 1, 5, 1, 2, 5, 6, 1, 3, 5, 3, 1, 2, 4, 1, 3, 5, 1, 2, 3, 4, 5, 6, 1, 5, 7, 1, 2, 5, 6, 7, 8, 1, 3, 7, 9, 1, 2, 4, 8, 10, 1, 3, 5, 9, 5, 1, 2, 3, 4, 5, 6, 9, 10, 1, 5, 7, 11, 1, 2, 6, 7, 8, 11, 12, 1, 3, 3, 7, 4, 9, 11

Offset: 1

Author

Glen Whitney, Feb 11 2023

Keywords

• frac
• nonn
• tabf

Comments

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 numerator of the n-th rational number (in lowest terms) added to the tree.

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.

For a proof that the tree described above includes all rational numbers between 0 and 1, see Gordon and Whitney.

Examples

			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 1, 1, 1, 2, 1, ... (the numerators of 1/2, 1/3, 1/4, 2/3, 1/5, ...).
		

Links

• Glen Whitney, Table of n, a(n) for n = 1..10052
• G. Gordon and G. Whitney, The Playground Problem 367, Math Horizons, Vol. 26 No. 1 (2018), 32-33.

Crossrefs

Denominators in A360565.

Level sizes of the tree in A360566.

See also the Farey tree in A007305 and A007306.

Cf. A293247.

Programs

Python

from fractions import Fraction
row = [Fraction(1,2)]
seen = set([Fraction(1,2), Fraction(1,1)])
limit = 25 # chosen to generate 10000 fractions
nums = []
denoms = []
rowsizes = []
while row[0].denominator <= limit:
   rowsizes.append(len(row))
   newrow = []
   for frac in row:
      nums.append(frac.numerator)
      denoms.append(frac.denominator)
      for nf in [Fraction(frac.numerator, frac.denominator+1),
                 Fraction(frac.numerator+1, frac.denominator)]:
         if not(nf in seen):
            newrow.append(nf)
            seen.add(nf)
   row = newrow
print(', '.join(str(num) for num in nums))
print(', '.join(str(denom) for denom in denoms))
print(', '.join(str(rowsize) for rowsize in rowsizes))