%I #8 Oct 13 2012 04:01:49
%S 0,0,0,0,0,0,0,0,2,14,62,235,821,2868,10193,36404,130174,466913,
%T 1681999,6083873
%N Number of perfect squared rectangles of order n up to symmetries of the rectangle.
%C A squared rectangle (which may be a square) is a rectangle dissected into a finite number, two or more, of squares. If no two of these squares have the same size the squared rectangle is perfect. The order of a squared rectangle is the number of constituent squares.
%C A squared rectangle is simple if it does not contain a smaller squared rectangle, compound if it does, and trivially compound if a constituent square has the same side length as a side of the squared rectangle under consideration.
%D See crossrefs for references and links.
%H <a href="/index/Sq#squared_rectangles">Index entries for squared rectangles</a>
%H <a href="/index/Sq#squared_squares">Index entries for squared squares</a>
%F a(n) = A002839(n) + A217153(n) + A217375(n).
%F a(n) >= 2*a(n-1) + A002839(n) + 2*A002839(n-1) + A217153(n) + 2*A217153(n-1), with equality for n<19.
%e a(10) = 14 comprises the A002839(10) = 6 simple perfect squared rectangles (SPSRs) of order 10 and the 8 trivially compound perfect squared rectangles which each comprises one of the two order 9 SPSRs and one other square.
%Y Cf. A110148 (counts symmetries of any squared subrectangles as equivalent).
%Y Cf. A181735, A217156.
%K nonn,hard,more
%O 1,9
%A _Geoffrey H. Morley_, Sep 27 2012
%E a(19) and a(20) corrected by _Geoffrey H. Morley_, Oct 12 2012