Remove duplicate implementation of Collatz sequence (#8836)

* updating DIRECTORY.md

* Remove duplicate implementation of Collatz sequence

* updating DIRECTORY.md

* Add suggestions from PR review

---------

Co-authored-by: github-actions <${GITHUB_ACTOR}@users.noreply.github.com>

maths/collatz_sequence.py

@@ -1,43 +1,66 @@
+"""
+The Collatz conjecture is a famous unsolved problem in mathematics. Given a starting
+positive integer, define the following sequence:
+- If the current term n is even, then the next term is n/2.
+- If the current term n is odd, then the next term is 3n + 1.
+The conjecture claims that this sequence will always reach 1 for any starting number.
+
+Other names for this problem include the 3n + 1 problem, the Ulam conjecture, Kakutani's
+problem, the Thwaites conjecture, Hasse's algorithm, the Syracuse problem, and the
+hailstone sequence.
+
+Reference: https://en.wikipedia.org/wiki/Collatz_conjecture
+"""
+
 from __future__ import annotations
 
+from collections.abc import Generator
 
-def collatz_sequence(n: int) -> list[int]:
+
+def collatz_sequence(n: int) -> Generator[int, None, None]:
     """
-    Collatz conjecture: start with any positive integer n. The next term is
-    obtained as follows:
-        If n term is even, the next term is: n / 2 .
-        If n is odd, the next term is: 3 * n + 1.
-
-    The conjecture states the sequence will always reach 1 for any starting value n.
-    Example:
-    >>> collatz_sequence(2.1)
+    Generate the Collatz sequence starting at n.
+    >>> tuple(collatz_sequence(2.1))
     Traceback (most recent call last):
         ...
-    Exception: Sequence only defined for natural numbers
-    >>> collatz_sequence(0)
+    Exception: Sequence only defined for positive integers
+    >>> tuple(collatz_sequence(0))
     Traceback (most recent call last):
         ...
-    Exception: Sequence only defined for natural numbers
-    >>> collatz_sequence(43)  # doctest: +NORMALIZE_WHITESPACE
-    [43, 130, 65, 196, 98, 49, 148, 74, 37, 112, 56, 28, 14, 7,
-     22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1]
+    Exception: Sequence only defined for positive integers
+    >>> tuple(collatz_sequence(4))
+    (4, 2, 1)
+    >>> tuple(collatz_sequence(11))
+    (11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1)
+    >>> tuple(collatz_sequence(31))     # doctest: +NORMALIZE_WHITESPACE
+    (31, 94, 47, 142, 71, 214, 107, 322, 161, 484, 242, 121, 364, 182, 91, 274, 137,
+    412, 206, 103, 310, 155, 466, 233, 700, 350, 175, 526, 263, 790, 395, 1186, 593,
+    1780, 890, 445, 1336, 668, 334, 167, 502, 251, 754, 377, 1132, 566, 283, 850, 425,
+    1276, 638, 319, 958, 479, 1438, 719, 2158, 1079, 3238, 1619, 4858, 2429, 7288, 3644,
+    1822, 911, 2734, 1367, 4102, 2051, 6154, 3077, 9232, 4616, 2308, 1154, 577, 1732,
+    866, 433, 1300, 650, 325, 976, 488, 244, 122, 61, 184, 92, 46, 23, 70, 35, 106, 53,
+    160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1)
+    >>> tuple(collatz_sequence(43))     # doctest: +NORMALIZE_WHITESPACE
+    (43, 130, 65, 196, 98, 49, 148, 74, 37, 112, 56, 28, 14, 7, 22, 11, 34, 17, 52, 26,
+    13, 40, 20, 10, 5, 16, 8, 4, 2, 1)
     """
-
     if not isinstance(n, int) or n < 1:
-        raise Exception("Sequence only defined for natural numbers")
+        raise Exception("Sequence only defined for positive integers")
 
-    sequence = [n]
+    yield n
     while n != 1:
-        n = 3 * n + 1 if n & 1 else n // 2
-        sequence.append(n)
-    return sequence
+        if n % 2 == 0:
+            n //= 2
+        else:
+            n = 3 * n + 1
+        yield n
 
 
 def main():
     n = 43
-    sequence = collatz_sequence(n)
+    sequence = tuple(collatz_sequence(n))
     print(sequence)
-    print(f"collatz sequence from {n} took {len(sequence)} steps.")
+    print(f"Collatz sequence from {n} took {len(sequence)} steps.")
 
 
 if __name__ == "__main__":