Centered Polygonal Number

The centered polygonal numbers are a family of sequences of 2-dimensional figurate numbers, each formed by a central dot, surrounded by polygonal layers with a constant number of sides. Each side of a polygonal layer contains one dot more than a side in the previous layer.

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• | - Centered triangular numbers | Centered square numbers | Centered pentagonal numbers | Centered hexagonal numbers

In the following table are listed the first 12 terms of the sequences of centered k-polygonal numbers, with k from 3 to 22:

As you can see:

• 6 is the 1st pentagonal number.
• 16 is the 2nd pentagonal number and the 1st pentadecagonal number.
• 19 is the 3rd triangular number, the 2nd hexagonal number and the 1st octadecagonal number.

Write a function that takes an integer n as argument and returns its classification as polygonal number:

• return "0th of all" if n is 1, since it is the 0th term of every centered polygonal number sequence.
• return a list whose generic element is a string formatted as "{i}th {k}-gonal number" if n is the i-th k-gonal number, in k-ascending order.
• return False if n is not a k-gonal number for any k > 2.

Examples

is_polygonal(3) ➞ False

is_polygonal(4) ➞ ["1st 3-gonal number"]

is_polygonal(16) ➞ ["2nd 5-gonal number", "1st 15-gonal number"]

Notes

N/A