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

1,3

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.

References

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

Links

Table of n, a(n) for n=1..13.

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.

Example

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.

Crossrefs

Cf. A004003.

Sequence in context: A262123 A005749 A005739 * A226588 A367279 A318641

Adjacent sequences: A279884 A279885 A279886 * A279888 A279889 A279890

Keyword

nonn,more

Author

Greg Huber, Dec 21 2016

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

Status

approved