A279887 Number of tilings of a sphinx of order n by elementary sphinxes (i.e., sphinxes of order 1).

1, 1, 4, 16, 153, 71838, 5965398, 2614508085, 9822629511079, 28751930151895611, 162231215752303027270, 32813942272624544838651213, 1257159787425487037702548758466

Offset: 1

Author

Greg Huber, Dec 21 2016

Keywords

• nonn
• more

Comments

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.

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).

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

a(10), a(11) from double hemisphinx method described above.

Examples

			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.
		

References

• A. Martin, "The Sphinx Task Centre Problem" in C. Pritchard (ed.) The Changing Shape of Geometry, Cambridge Univ. Press, 2003, 371-378.

Links

• Greg Huber, Craig Knecht, Walter Trump, and Robert M. Ziff, Riddles of the sphinx tilings, arXiv:2304.14388 [cond-mat.stat-mech], 2023.
• Greg Huber, Craig Knecht, Walter Trump, and Robert M. Ziff, Entropy and chirality in sphinx tilings, Phys. Rev. Res., 6 (2024), 013227.
• J.-Y. Lee and R. V. Moody, Lattice Substitution Systems and Model Sets, arXiv:math/0002019 [math.MG], 2000.
• J.-Y. Lee and R. V. Moody, Lattice Substitution Systems and Model Sets, Discrete Comput. Geom., 25 (2001), 173-201.
• Mathematics Task Centre, Task166.
• Walter Trump, The Dangler Method
• University of Bielefeld Tilings, Sphinx.
• Wikipedia, Sphinx tiling.
• Wikiwand, Sphinx Tiling.

Crossrefs

Cf. A004003.

Extensions

a(9) from Greg Huber, Mar 10 2017

a(10)-a(11) from Greg Huber, May 10 2017

a(11) corrected by Walter Trump, Feb 25 2022

a(12)-a(13) from Walter Trump, Feb 25 2022