Stern-Brocot sequence

Stern-Brocot sequence is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

For this task, the Stern-Brocot sequence is to be generated by an algorithm similar to that employed in generating the Fibonacci sequence.

1. The first and second members of the sequence are both 1
    1, 1
2. Start by considering the second member of the sequence
3. Sum the considered member of the sequence and its precedent, (1 + 1) = 2, and append it to the end of the sequence.
    1, 1, 2
4. Append the considered member of the sequence to the end of the sequence.
    1, 1, 2, 1
5. Consider the next member of the series, (the third member i.e. 2)
6. GOTO 3

Expanding another loop we get:

7. Sum the considered member of the sequence and its precedent, (2 + 1) = 3, and append it to the end of the sequence.
    1, 1, 2, 1, 3
8. Append the considered member of the sequence to the end of the sequence.
    1, 1, 2, 1, 3, 2
9. Consider the next member of the series, (the fourth member i.e. 1)

The task is to

1. Create a function/method/subroutine/procedure/... to generate the Stern-Brocot sequence of integers.
2. Show the first fifteen members of the sequence. (This should be: 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4)
3. Show the (1-based) index of where the numbers 1-to-10 first appears in the sequence.
4. Show the (1-based) index of where the number 100 first appears in the sequence.
5. Check that the greatest common divisor of all the two consecutive members of the series up to the 1000th member, is always one.

Show your output on the page.

Ref

• Infinite Fractions - Numberphile (Video).
• Trees, Teeth, and Time: The mathematics of clock making.
• A002487 The On-Line Encyclopedia of Integer Sequences.

Python

<lang python>def stern_brocot(predicate=lambda series: len(series) < 20):

   """\
   Generates members of the stern-brocot series, in order, returning them when the predicate becomes false

   >>> print('The first 10 values:',
             stern_brocot(lambda series: len(series) < 10)[:10])
   The first 10 values: [1, 1, 2, 1, 3, 2, 3, 1, 4, 3]
   >>>
   """

   sb, i = [1, 1], 0
   while predicate(sb):
       sb += [sum(sb[i:i + 2]), sb[i + 1]]
       i += 1
   return sb

if __name__ == '__main__':

   from fractions import gcd

   n_first = 15
   print('The first %i values:\n  ' % n_first,
         stern_brocot(lambda series: len(series) < n_first)[:n_first])
   print()
   n_max = 10
   for n_occur in list(range(1, n_max + 1)) + [100]:
       print('1-based index of the first occurrence of %3i in the series:' % n_occur,
             stern_brocot(lambda series: n_occur not in series).index(n_occur) + 1)
   print()
   n_gcd = 1000
   s = stern_brocot(lambda series: len(series) < n_gcd)[:n_gcd]
   assert all(gcd(prev, this) == 1
              for prev, this in zip(s, s[1:])), 'A fraction from adjacent terms is reducible'</lang>

Output:

The first 15 values:
   [1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4]

1-based index of the first occurrence of   1 in the series: 1
1-based index of the first occurrence of   2 in the series: 3
1-based index of the first occurrence of   3 in the series: 5
1-based index of the first occurrence of   4 in the series: 9
1-based index of the first occurrence of   5 in the series: 11
1-based index of the first occurrence of   6 in the series: 33
1-based index of the first occurrence of   7 in the series: 19
1-based index of the first occurrence of   8 in the series: 21
1-based index of the first occurrence of   9 in the series: 35
1-based index of the first occurrence of  10 in the series: 39
1-based index of the first occurrence of 100 in the series: 1179