[seqfan] Packing many different square sizes in a square

[Allan Wechsler]

I thought that all the different changes on packing squares into a square
had been rung, but apparently it is not so.

Take a square of integer side n > 0. Let A(n) be the largest number of
different sizes that can be incorporated in a tiling of the big square by
smaller integer squares.

Up to n=16, the answer seems to be ceiling(n/2), but I cannot pack 9
different square sizes on a 17x17 board.  The sequence I have so far is

1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,8,9,9,9

and if this is right, it's not in OEIS yet. Can any sequence fanatic
increase any of these numbers?  Or give more terms? (Or better yet, write
code to search for tilings that maximize the number of square sizes?) I'll
submit this soon unless somebody can show that I've erred. Thanks in
advance!