Added nth_sgonal_num.py (#8753)

* Added nth_sgonal_num.py

* Update and rename nth_sgonal_num.py to polygonal_numbers.py

---------

Co-authored-by: Tianyi Zheng <tianyizheng02@gmail.com>

maths/polygonal_numbers.py

@@ -0,0 +1,32 @@
+def polygonal_num(num: int, sides: int) -> int:
+    """
+    Returns the `num`th `sides`-gonal number. It is assumed that `num` >= 0 and
+    `sides` >= 3 (see for reference https://en.wikipedia.org/wiki/Polygonal_number).
+
+    >>> polygonal_num(0, 3)
+    0
+    >>> polygonal_num(3, 3)
+    6
+    >>> polygonal_num(5, 4)
+    25
+    >>> polygonal_num(2, 5)
+    5
+    >>> polygonal_num(-1, 0)
+    Traceback (most recent call last):
+        ...
+    ValueError: Invalid input: num must be >= 0 and sides must be >= 3.
+    >>> polygonal_num(0, 2)
+    Traceback (most recent call last):
+        ...
+    ValueError: Invalid input: num must be >= 0 and sides must be >= 3.
+    """
+    if num < 0 or sides < 3:
+        raise ValueError("Invalid input: num must be >= 0 and sides must be >= 3.")
+
+    return ((sides - 2) * num**2 - (sides - 4) * num) // 2
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()