Small areas within the sphinx that are capable of multiple tilings are important drivers of the total enumeration.
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.
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.
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.
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