The P-pentomino is rep-n² for all n. The diagram below shows solutions for n = 2, 3, 5 and a method of extending the rep-5² to rep-7², rep-9², rep11² and rep13² which can be extended to give solutions for all odd n. Since any even n is composite this shows that the P is rep-n² for all n.
The diagram below shows how a rep-n² can be extended to a rep-(n+5)².
The Y-pentomino forms 10x10 and 15x15 squares and so is rep-(10k)² and rep-(15k)². Mike Reid reports that it is also rep-n² for n = 9, 11 and 16. These are shown below - the rep-16² and the rep-19² use a number of Y-pentomino rectangles - details of these can be found in Michael Reid's Rectifiable polyomino page or Torsten Sillke's Y pentomino page.
Below are general constructions for rep-(5n)² and rep-(5n+1)². (If m>2 then a 5m x k rectangle exists for all k > 19 and so all the rectangles exist for n > 4- see possible rectangles) This shows that the Y-pentomino is rep-n² for all n = 0, 1 (mod 5).
The next image shows how to create a rep-54 from three rep-9 and three rectangles.
The constructions below, made from I- and Y-pentominoes, can be expanded by a factor of 9 to give solutions to the rep-63 and rep-72 using the rep-9 for each Y and the 9x45 rectangle for each I. Also, since an 11x55 rectangle exists for the Y and a rep-11 is possible, we can also construct a rep-77.
The next diagram shows how a rep-n can be extended by 10 (or 15) provided a 10xn (or 15xn) rectangle exists.
Mike Reid has also found the following replications. Each has a part made up of Y-pentominoes and the figures are then completed with various rectangles which can be made with the Y-pentominoes.
The above shows that the Y is rep-n for all n>13. There is no rep-n for n = 2,3,4,5,6,7,8 and so the only doubtful case is now rep-12
Patricvk Hamlyn has now found the rep-12 as shown below.
The above are the only pentominoes which are reptiles but we could, as with rectifiable polyominoes, consider pairs of pentominoes. The problem here would be to create n-fold copies of the pair of pentominoes. Three examples are given below - V and Z (rep-5); F and U (rep-6); U and N (rep-7); and T and W (rep-9).