

A279887


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


4



1, 1, 4, 16, 153, 71838, 5965398, 2614508085, 9822629511079, 28751930151895611, 155212395372255675054
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OFFSET

1,3


COMMENTS

Sphinx tilings are, by convention, understood to be improper tilings composed of two elementary shapes, order1 sphinxes, that are mirror images of one another. In other words, one can prove that the tiling of an ordern sphinx requires both Lsphinxes and Rsphinxes (each composed of six equilateral triangles) for any n>1. The sequence terms are based on an initial searchtree method by G. Huber, confirmed and extended by W. Trump using backtracking and a bitvector method.
Leastsquares 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 (W. Trump, personal communication).  Greg Huber, Mar 10 2017
a(10), a(11) from double hemisphinx method described above.


REFERENCES

G. Huber, C. Knecht, W. Trump, and R. M. Ziff, "The Riddle of the Sphinx", 2016, unpublished.
A. Martin, "The Sphinx Task Centre Problem" in C. Pritchard (ed.) The Changing Shape of Geometry, Cambridge Univ. Press, 2003, 371378.


LINKS

Table of n, a(n) for n=1..11.
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), 173201.
Mathematics Task Centre, Task166.
University of Bielefeld Tilings, Sphinx.
Wikipedia, Sphinx tiling.
Wikiwand, Sphinx Tiling.


EXAMPLE

For n=2, a(2)=1 and this single tiling of an order2 Lsphinx with three elementary Rsphinxes and one elementary Lsphinx is shown in the Wikiwand link.


CROSSREFS

Cf. A004003.
Sequence in context: A262123 A005749 A005739 * A226588 A318641 A005741
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


STATUS

approved



