cp's OEIS Frontend

A108066 Number of distinct ways to dissect a square into n rectangles of equal area.

%I A108066 #46 Feb 16 2025 08:32:58
%S A108066 1,1,2,6,18,65,281,1343,6953,38023
%N A108066 Number of distinct ways to dissect a square into n rectangles of equal area.
%C A108066 "Distinct" here means that dissections differing only by a rotation and/or reflection are not counted as different (see A189243).
%C A108066 The first time the pieces can be made to all have different shapes (but the same area) is at n=7 - see Descartes (1971) and the illustration; also Wells, Weisstein. - _N. J. A. Sloane_, Dec 05 2012
%D A108066 David Wells, Penguin Dictionary of Curious and Interesting Geometry, 1991, pp. 15-16.
%H A108066 Blanche Descartes, <a href="https://archim.org.uk/eureka/archive/Eureka-34.pdf">Divisions of a square into rectangles</a>, Eureka, No. 34 (1971), 31-35.
%H A108066 R. Häggkvist, P.-O. Lindberg, and B. Lindström, <a href="http://dx.doi.org/10.1016/0012-365X(83)90103-6">Dissecting a square into rectangles of equal area</a>, Discr. Math. 47 (1983), 321-323.
%H A108066 Johan Nilsson, <a href="/A108066/a108066.pdf">Illustrations for terms up to a(6).</a>.
%H A108066 N. J. A. Sloane, <a href="/A108066/a108066.jpg">Blanche Descartes's dissection into 7 pieces with equal areas but different shapes</a>.
%H A108066 Eric Weisstein, <a href="https://mathworld.wolfram.com/BlanchesDissection.html">Blanche's Dissection</a>.
%e A108066 There are six ways to dissect a square into four rectangles of equal area, so a(4)=6:
%e A108066 +-+-----+ +-+-+---+ +-+-----+ +-+-+-+-+ +-+---+-+ +---+---+
%e A108066 | |     | | | |   | | |     | | | | | | | |   | | |   |   |
%e A108066 | |     | | | |   | | |     | | | | | | | |   | | |   |   |
%e A108066 | +--+--+ | | |   | | +-----+ | | | | | | |   | | |   |   |
%e A108066 | |  |  | | | +---+ | |     | | | | | | | +---+ | +---+---+
%e A108066 | |  |  | | | |   | | |_____| | | | | | | |   | | |   |   |
%e A108066 | |  |  | | | |   | | |     | | | | | | | |   | | |   |   |
%e A108066 | |  |  | | | |   | | |     | | | | | | | |   | | |   |   |
%e A108066 +-+--+--+ +-+-+---+ +-+-----+ +-+-+-+-+ +-+---+-+ +---+---+
%Y A108066 Cf. A189243, A219861, A359146.
%K A108066 hard,more,nonn,nice
%O A108066 1,3
%A A108066 Hans Riesebos (hans.riesebos(AT)wanadoo.nl) and _Herman Beeksma_, Jun 03 2005