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%I #59 Sep 15 2018 15:55:20
%S 0,0,0,5,18,48170,8361983
%N Number of tilings of a sphinx of order n with n^2 - 2 elementary sphinxes and a single sphinx domino that has two different tilings.
%C Small areas within the sphinx that are capable of multiple tilings are important drivers of the total enumeration.
%C The smallest area that can have two different tilings with the elementary sphinx is a sphinx domino. This unique domino is replaced with a single tile defect for this sequence. This domino is called a flacon.
%C This replacement causes fewer tilings for sphinxes of orders six and below and more tilings for the order seven sphinx when compared to a pure sphinx tiling A279887. Figuring out why that happens makes this sequence interesting.
%C The 153 order 5 pure sphinx tilings are shown in the links below. The 12 tile aspects are color coded. The blacked out areas show the tiles that change from tiling a(n) to a(n+1). Tilings #4 and #13 show the smallest areas that have two different tilings. Tilings # 63 and # 64 show that all sphinx tiles will change position in going through the 153 examples. This particular listing has tiling pairs that always share 2 or more sphinx tiles that do not change position. The sphinx tiles that change position are always edge joined.
%C Combining the 12 aspects of the sphinx tile produces 46 sphinx dominoes. Sphinx domino tiling is compared with sphinx tiling in the order 4 sphinx (see link below). - _Craig Knecht_, Sep 08 2018
%H Craig Knecht, <a href="/A317541/a317541_17.png">Domino sphinx tiling.</a>
%H Craig Knecht, <a href="/A317541/a317541_4.png">Sequence example.</a>
%H Craig Knecht, <a href="/A317541/a317541_16.png">Order 4 Sphinx.</a>
%H Craig Knecht, <a href="/A317541/a317541_10.png">Order 5 sphinx 1 to 70.</a>
%H Craig Knecht, <a href="/A317541/a317541_13.png">Order 5 sphinx 71 to 130.</a>
%H Craig Knecht, <a href="/A317541/a317541_14.png">Order 5 sphinx 131 to 153.</a>
%H Craig Knecht, <a href="/A317541/a317541_6.png">Order 6 sphinx containing 7 flacons.</a>
%H Craig Knecht, <a href="/A317541/a317541_3.png">Order 7 sphinx containing 12 flacons.</a>
%H Craig Knecht, <a href="/A317541/a317541_5.png">Order 7 sphinx five sphinx tile embedded shapes.</a>
%H Craig Knecht, <a href="/A317541/a317541_9.png">Small symmetric sphinx tilings.</a>
%H Craig Knecht, <a href="/A317541/a317541_15.png">Sphinx tile basics.</a>
%Y Cf. A279887.
%K nonn,more
%O 0,4
%A _Craig Knecht_, Jul 30 2018
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