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%I #74 Apr 04 2024 12:46:03
%S 1,1,4,16,153,71838,5965398,2614508085,9822629511079,
%T 28751930151895611,162231215752303027270,32813942272624544838651213,
%U 1257159787425487037702548758466
%N Number of tilings of a sphinx of order n by elementary sphinxes (i.e., sphinxes of order 1).
%C Sphinx tilings are, by convention, understood to be improper tilings composed of two elementary shapes, order-1 sphinxes, that are mirror images of one another. In other words, one can prove that the tiling of an order-n sphinx requires both L-sphinxes and R-sphinxes (each composed of six equilateral triangles) for any n>1. The sequence terms are based on an initial search-tree method by G. Huber, confirmed and extended by _Walter Trump_ using backtracking and a bit-vector method.
%C Least-squares fitting indicates a growth law in the form of an exponential of a quadratic in n (i.e., proportional to g^(area), where g is a constant).
%C a(9) from analysis of the tilings and associated seam factor of two hemisphinxes of order 9 (_Walter Trump_, personal communication). - _Greg Huber_, Mar 10 2017
%C a(10), a(11) from double hemisphinx method described above.
%D A. Martin, "The Sphinx Task Centre Problem" in C. Pritchard (ed.) The Changing Shape of Geometry, Cambridge Univ. Press, 2003, 371-378.
%H Greg Huber, Craig Knecht, Walter Trump, and Robert M. Ziff, <a href="https://arxiv.org/abs/2304.14388">Riddles of the sphinx tilings</a>, arXiv:2304.14388 [cond-mat.stat-mech], 2023.
%H Greg Huber, Craig Knecht, Walter Trump, and Robert M. Ziff, <a href="http://dx.doi.org/10.1103/PhysRevResearch.6.013227">Entropy and chirality in sphinx tilings</a>, Phys. Rev. Res., 6 (2024), 013227.
%H J.-Y. Lee and R. V. Moody, <a href="https://arxiv.org/abs/math/0002019">Lattice Substitution Systems and Model Sets</a>, arXiv:math/0002019 [math.MG], 2000.
%H J.-Y. Lee and R. V. Moody, <a href="https://doi.org/10.1007/s004540010083">Lattice Substitution Systems and Model Sets</a>, Discrete Comput. Geom., 25 (2001), 173-201.
%H Mathematics Task Centre, <a href="http://www.mathematicscentre.com/taskcentre/166sfinx.htm">Task166</a>.
%H Walter Trump, <a href="/A279887/a279887.pdf">The Dangler Method</a>
%H University of Bielefeld Tilings, <a href="http://tilings.math.uni-bielefeld.de/substitution/sphinx/">Sphinx</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Sphinx_tiling">Sphinx tiling</a>.
%H Wikiwand, <a href="http://www.wikiwand.com/en/Sphinx_tiling">Sphinx Tiling</a>.
%e For n=2, a(2)=1 and this single tiling of an order-2 L-sphinx with three elementary R-sphinxes and one elementary L-sphinx is shown in the Wikiwand link.
%Y Cf. A004003.
%K nonn,more
%O 1,3
%A _Greg Huber_, Dec 21 2016
%E a(9) from _Greg Huber_, Mar 10 2017
%E a(10)-a(11) from _Greg Huber_, May 10 2017
%E a(11) corrected by _Walter Trump_, Feb 25 2022
%E a(12)-a(13) from _Walter Trump_, Feb 25 2022
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