Add Sylvester's sequence to maths (#5171)

* Add Sylvester's sequence to maths

* Update sylvester_sequence.py

Co-authored-by: Christian Clauss <cclauss@me.com>

maths/sylvester_sequence.py

@@ -0,0 +1,43 @@
+"""
+
+Calculates the nth number in Sylvester's sequence
+
+Source:
+    https://en.wikipedia.org/wiki/Sylvester%27s_sequence
+
+"""
+
+
+def sylvester(number: int) -> int:
+    """
+    :param number: nth number to calculate in the sequence
+    :return: the nth number in Sylvester's sequence
+
+    >>> sylvester(8)
+    113423713055421844361000443
+
+    >>> sylvester(-1)
+    Traceback (most recent call last):
+    ...
+    ValueError: The input value of [n=-1] has to be > 0
+
+    >>> sylvester(8.0)
+    Traceback (most recent call last):
+    ...
+    AssertionError: The input value of [n=8.0] is not an integer
+    """
+    assert isinstance(number, int), f"The input value of [n={number}] is not an integer"
+
+    if number == 1:
+        return 2
+    elif number < 1:
+        raise ValueError(f"The input value of [n={number}] has to be > 0")
+    else:
+        num = sylvester(number - 1)
+        lower = num - 1
+        upper = num
+        return lower * upper + 1
+
+
+if __name__ == "__main__":
+    print(f"The 8th number in Sylvester's sequence: {sylvester(8)}")