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%I #10 Oct 13 2012 04:00:51
%S 0,0,0,0,0,0,0,0,2,10,38,127,408,1375,4783,16645,58059,203808,722575
%N Number of perfect squared rectangles of order n up to symmetries of the rectangle and of its subrectangles if any.
%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. [_Geoffrey H. Morley_, Oct 12 2012]
%H C. J. Bouwkamp, On the dissection of rectangles into squares (Papers I-III), Koninklijke Nederlandsche Akademie van Wetenschappen, Proc., Ser. A, <a href="http://www.dwc.knaw.nl/DL/publications/PU00018283.pdf">Paper I</a>, 49 (1946), 1176-1188 (=Indagationes Math., v. 8 (1946), 724-736); <a href="http://www.dwc.knaw.nl/DL/publications/PU00018294.pdf">Paper II</a>, 50 (1947), 58-71 (=Indagationes Math., v. 9 (1947), 43-56); <a href="http://www.dwc.knaw.nl/DL/publications/PU00018295.pdf">Paper III</a>, 50 (1947), 72-78 (=Indagationes Math., v. 9 (1947), 57-63). [Paper I has terms up to a(12) and an incorrect value for a(13) on p. 1178.]
%H C. J. Bouwkamp, <a href="http://www.dwc.knaw.nl/DL/publications/PU00018444.pdf">On the construction of simple perfect squared squares</a>, Koninklijke Nederlandsche Akademie van Wetenschappen, Proc., Ser. A, 50 (1947), 1296-1299 (=Indagationes Math., v. 9 (1947), 622-625). [Correct terms up to a(13) on p. 1299.]
%H I. M. Yaglom, <a href="http://ilib.mirror1.mccme.ru/djvu/yaglom/square.htm">How to dissect a square?</a> (in Russian), Nauka, Moscow, 1968. In djvu format (1.7M), also as this <a href="http://www.squaring.net/downloads/Yaglom-square.pdf">pdf</a> (9.5M). [Terms up to a(13) on pp. 26-7.]
%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) + A217152(n) + A217374(n). - _Geoffrey H. Morley_, Oct 12 2012
%F a(n) = a(n-1) + A002839(n) + A002839(n-1) + A217152(n) + A217152(n-1). - _Geoffrey H. Morley_, Oct 12 2012
%Y Cf. A217154 (counts symmetries of any subrectangles as distinct).
%Y Cf. A181735, A217153, A217156.
%K nonn,hard,more
%O 1,9
%A Tanya Khovanova, Feb 18 2007
%E Definition corrected and a(14)-a(19) added by _Geoffrey H. Morley_, Oct 12 2012
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