Add Zeckendorf representation algorithm

Implements Zeckendorf's theorem: every positive integer can be written
uniquely as the sum of one or more distinct, non-consecutive Fibonacci
numbers. The representation is recovered with a greedy pass from the
largest Fibonacci number downwards, which is optimal for this problem.

Includes doctests for 0, 1, 2, 4, 10, 100 and 1000, plus TypeError for
non-integers and ValueError for negative input.

https://en.wikipedia.org/wiki/Zeckendorf%27s_theorem

Author: mgkvibez

maths/zeckendorf.py

@@ -0,0 +1,76 @@
+"""
+https://en.wikipedia.org/wiki/Zeckendorf%27s_theorem
+"""
+
+
+def _fibonacci_numbers_up_to(limit: int) -> list[int]:
+    """
+    Return the Fibonacci numbers (1, 2, 3, 5, ...) that are <= limit.
+
+    The sequence starts at 1, 2 rather than 1, 1 so that every number has a
+    unique representation and the value 1 is used at most once.
+
+    >>> _fibonacci_numbers_up_to(1)
+    [1]
+    >>> _fibonacci_numbers_up_to(10)
+    [1, 2, 3, 5, 8]
+    >>> _fibonacci_numbers_up_to(0)
+    []
+    """
+    numbers: list[int] = [1, 2]
+    while numbers[-1] + numbers[-2] <= limit:
+        numbers.append(numbers[-1] + numbers[-2])
+    return [number for number in numbers if number <= limit]
+
+
+def zeckendorf(number: int) -> list[int]:
+    """
+    Return the Zeckendorf representation of a non-negative integer.
+
+    Zeckendorf's theorem states that every positive integer can be written
+    uniquely as the sum of one or more distinct, non-consecutive Fibonacci
+    numbers. This function recovers that representation with a greedy pass
+    over the Fibonacci numbers from largest to smallest, which is always
+    optimal for this problem.
+
+    >>> zeckendorf(0)
+    []
+    >>> zeckendorf(1)
+    [1]
+    >>> zeckendorf(2)
+    [2]
+    >>> zeckendorf(4)
+    [3, 1]
+    >>> zeckendorf(10)
+    [8, 2]
+    >>> zeckendorf(100)
+    [89, 8, 3]
+    >>> zeckendorf(1000)
+    [987, 13]
+    >>> zeckendorf(-1)
+    Traceback (most recent call last):
+        ...
+    ValueError: zeckendorf() only accepts non-negative integers
+    >>> zeckendorf(1.5)
+    Traceback (most recent call last):
+        ...
+    TypeError: zeckendorf() only accepts integers
+    """
+    if isinstance(number, bool) or not isinstance(number, int):
+        raise TypeError("zeckendorf() only accepts integers")
+    if number < 0:
+        raise ValueError("zeckendorf() only accepts non-negative integers")
+
+    representation: list[int] = []
+    remainder = number
+    for fibonacci_number in reversed(_fibonacci_numbers_up_to(number)):
+        if fibonacci_number <= remainder:
+            representation.append(fibonacci_number)
+            remainder -= fibonacci_number
+    return representation
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()