Squared squares and squared rectangles are called simple if they do not contain a smaller squared square or rectangle. Simple perfect squared rectangles begin at order 9.
Squared squares and squared rectangles are called perfect if the squares in the tiling are all of different sizes.
Simple perfect squared rectangles have been have been catalogued from order 9 to 17 in javascript menus. Initially arranged by order and then by width and height, but further selected catalogues arranged by other characteristics are planned.
The main method of classifying Squared Rectangles is to organise them with a given number of elements, by order, that is, the number of constituent squares, then by width then by height.
Catalogues have been produced of simple perfect squared rectangles(SPSR) in the search for squared squares.
The On-Line Encyclopedia of Integer Sequences lists sequence A219766; the Number of nonsquare simple perfect squared rectangles of order n up to symmetry as; 0, 0, 0, 0, 0, 0, 0, 0, 2, 6, 22, 67, 213, 744, 2 609, 9 016, 31 426, 110 381, 390 223, 1 383 905, 4 931 307, 17 633 769 , 63 301 415, 228 130 900. In Ian Gambini's thesis he provided SPSR counts up to order 24, his counts for order 23 and 24 are confirmed despite being shown with question marks (indicating some uncertainty) in his thesis. Gambini's counts and Anderson's counts agree to order 24. It is not practical to display catalogues above order 17 due to the sheer number of tilings involved, however links to zipped bouwkampcodes (and tablecodes) up to order 21 are provided for connoisseurs of simple perfect squared rectangles.
A square is a rectangle, so the correct count for Simple Perfect Squared Rectangles includes Perfect Squared Squares from order 21 onwards. In the OEIS, this is listed as sequence A002839 ; Number of simple perfect squared rectangles of order n up to symmetry. ; 0, 0, 0, 0, 0, 0, 0, 0, 2, 6, 22, 67, 213, 744, 2609, 9016, 31426, 110381, 390223, 1383905, 4931308, 17633773, 63301427, 228130926.
Explanation of bouwkampcode and tablecode
[1] brackets indicate the number of compound rectangles identified and removed from the processed bouwkampcodes of that order. All bouwkampcodes are sorted, unique, with imperfects, compounds, and squared squares filtered out.