The study of polygonal numbers (triangular, square, etc.) has a long and rich history. Similar, but lesser known, are the centered polygonal numbers which have not been as extensively studied. These sequences of figurate numbers contain a wealth of interesting and fun mathematics. As an example, we make connections between the polygonal and centered polygonal numbers to find a previously unknown family of integer sequences that describe those numbers that are simultaneously polygonal and centered polygonal for the same number of sides.