A279887 -id:A279887 - cp's OEIS frontend

A291582 Maximum number of 6 sphinx tile shapes in a sphinx tiled hexagon of order n.

30, 132, 306, 552, 870, 1260, 1722, 2256, 2862, 3540, 4290, 5112, 6006, 6972, 8010, 9120, 10302, 11556, 12882, 14280, 15750, 17292, 18906, 20592, 22350, 24180, 26082, 28056, 30102, 32220, 34410, 36672, 39006, 41412, 43890, 46440, 49062, 51756, 54522, 57360, 60270, 63252

Offset: 1

Craig Knecht, Aug 30 2017

The equilateral triangle composed of 144 smaller equilateral triangles is the smallest triangle that can be tiled with the sphinx. This triangle is used to form all orders of the hexagon.
Walter Trump enumerated all 830 sphinx tilings of this triangle and found six symmetrical examples one of which is used to produce this sequence.
Hyper-packing is a term that describes the ability of a shape to contain a greater area of subshapes than its own area by overlapping the subshapes. There are 864 unit triangles in the order 1 hexagon. 30 of the subshapes hyper-packed into this hexagon would contain 30x6x6 or 1080 unit triangles if summed individually.
The prime numbers cannot be described by a formula. Subsets of the primes such as the balanced primes are more formula friendly (see comments to puzzle 920 below). - Craig Knecht, Apr 19 2018

G. C. Greubel, Table of n, a(n) for n = 1..5000
Craig Knecht, Eight sphinx tile frame tessellations.
Craig Knecht, Example for the sequence.
Craig Knecht, Order 4 Hexagon with 246 subshapes.
Craig Knecht, Sphinx Tile Component Triangles.
Craig Knecht, Sphinx tiling of the triangle used to make the hexagon.
Craig Knecht, T12 symmetric sphinx tilings.
Carlos Rivera, Puzzle 920. An enigma related to A291582, Primes puzzles and problems connections.
Wikipedia, Walter Trump.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A016969, A049452.

Cf. A279887, A287999.

List([1..30], n -> 6*n*(6*n-1)); # G. C. Greubel, Dec 04 2018

[6*n*(6*n-1): n in [1..50]]; // Vincenzo Librandi, Sep 20 2017

seq(6*n*(6*n-1),n=1..100); # Robert Israel, Sep 19 2017

Array[6 # (6 # - 1) &, 42] (* Michael De Vlieger, Sep 19 2017 *)
CoefficientList[Series[2(15 + 21 x)/(1-x)^3,{x, 0, 50}], x] (* Vincenzo Librandi, Sep 20 2017 *)
CoefficientList[Series[6 E^x (5 + 17 x + 6 x^2), {x, 0, 50}], x]*
Table[n!, {n, 0, 50}] (* Stefano Spezia, Dec 07 2018 *)

a(n) = 6*n*(6*n-1); \\ Altug Alkan, Apr 08 2018

[6*n*(6*n-1) for n in (1..50)] # G. C. Greubel, Dec 04 2018

a(n) = 6*n*(6*n-1). - Walter Trump
G.f.: 2*x*(15+21*x)/(1-x)^3. - Vincenzo Librandi, Sep 20 2017
a(n) = 6*A049452(n) = 6*n*A016969(n-1). - Torlach Rush, Nov 28 2018
E.g.f.: 6*exp(x)*(5 + 17*x + 6*x^2). - Stefano Spezia, Dec 07 2018
a(n) = A016970(n-1) + A016969(n-1). - Torlach Rush, Dec 10 2018
From Amiram Eldar, Jul 30 2024: (Start)
Sum_{n>=1} 1/a(n) = log(2)/3 + log(3)/4 - sqrt(3)*Pi/12.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/6 - log(2)/6 - arccoth(sqrt(3))/sqrt(3). (End)

A287999 Number of tilings of a stackable 8 sphinx tile shape.

Original entry on oeis.org

2, 20, 190, 1896, 19950, 221720, 2596498, 31839772

Offset: 0

Craig Knecht, Jun 04 2017

Long narrow frames such as these are a good place to search for small symmetric sphinx tiled shapes.

Craig Knecht, Example for sequence.
Craig Knecht, Sphinx tile classification.
Craig Knecht, Visual scan

Cf. A279887.

A317541 Number of tilings of a sphinx of order n with n^2 - 2 elementary sphinxes and a single sphinx domino that has two different tilings.

Original entry on oeis.org

0, 0, 0, 5, 18, 48170, 8361983

Offset: 0

Craig Knecht, Jul 30 2018

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

Craig Knecht, Domino sphinx tiling.
Craig Knecht, Sequence example.
Craig Knecht, Order 4 Sphinx.
Craig Knecht, Order 5 sphinx 1 to 70.
Craig Knecht, Order 5 sphinx 71 to 130.
Craig Knecht, Order 5 sphinx 131 to 153.
Craig Knecht, Order 6 sphinx containing 7 flacons.
Craig Knecht, Order 7 sphinx containing 12 flacons.
Craig Knecht, Order 7 sphinx five sphinx tile embedded shapes.
Craig Knecht, Small symmetric sphinx tilings.
Craig Knecht, Sphinx tile basics.

Cf. A279887.

A318897 Number of tilings of the even-order sphinx with the two dominoes that form the second-order sphinx.

Original entry on oeis.org

1, 8, 5433, 28925040

Offset: 1

Craig Knecht, Sep 05 2018

There are 46 sphinx dominoes. The order 2 sphinx is composed of two different dominoes. These two dominoes are used to tile the even-order sphinx.

The orientation of the order 8 sphinx in the link below is essential for the bit-vector bottom-up search to efficiently find solutions. All order 8 solutions are found in a few minutes.

Craig Knecht, 46 sphinx dominoes.
Craig Knecht, Sphinx domino introduction.
Craig Knecht, Sphinx domino tiling.
Craig Knecht, T12 missing one of the fundamental two dominoes.

Cf. A279887, A317541.

A289941 Number of tilings of n interlocking sphinx tiled shape subunits.

Original entry on oeis.org

18, 444, 11700, 320640, 9012060, 257487624

Offset: 1

Craig Knecht, Jul 15 2017

The shape for this sequence attempts to find a connection between the unique geometry of the sphinx tile and a polymerization process.
The number of tilings of a stackable sphinx tiled shape has previously been described A287999.
In contrast to the stackable shape this shape is interlocking.
This shape has one female and three male parts.
The properties of this shape that inhibit different polymerizations are noted.
The various polymer configurations from a central nucleus are shown.

Craig Knecht, Example for this sequence.
Craig Knecht, Geometry of the sphinx applied to a polymerization process.
Craig Knecht, Other examples of interlocking shapes.

Cf. A067411, A279887, A287999.

A318778 Number of different positions that an elementary sphinx can occupy in a sphinx of order n.

Original entry on oeis.org

1, 28, 128, 300, 544, 860, 1248, 1708, 2240, 2844

Offset: 1

Craig Knecht, Sep 10 2018

Craig Knecht, 33 positions that a flacon occupies in a S4 sphinx - animated.
Craig Knecht, Bottom row surface tension.
Craig Knecht, Order Two Sphinx - 28 positions.
Craig Knecht, Order three sphinx - 128 positions.
Craig Knecht, Various shapes positioned in a order 4 sphinx.

Cf. A279887, A317541, A318897.

Conjectures from Colin Barker, Nov 13 2018: (Start)
G.f.: x*(1 + 25*x + 47*x^2 - x^3) / (1 - x)^3.
a(n) = 44 - 80*n + 36*n^2 for n>1.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>4.
(End)

A366704 Number of sphinx tilings of T(n+12) with a central T(n) defect where T(k) is an equilateral triangle with side length k.

Original entry on oeis.org

830, 216, 144, 13760, 396, 144, 185348, 576, 144, 3222390, 756, 144, 57614324, 936, 144, 1033400616, 1116, 144, 18543135720, 1296, 144

Offset: 0

Craig Knecht, Oct 17 2023

A sphinx polyad frame has at least two different sphinx tilings where each of the elementary sphinx tiles occupies a different position.
The frames in this sequence that have 144 sphinx tilings led to the discovery of an infinite series of sphinx polyad frames.
How many polyiamonds can form an infinite series of fundamental polyads?

Eurekaalert, Riddles of the sphinx, 2024.
Greg Huber, Craig Knecht, Walter Trump, and Robert M. Ziff, Riddles of the sphinx tilings, arXiv:2304.14388 [cond-mat.stat-mech], 2023.
Craig Knecht, Chiral color coding of the sphinx tiles in the fundamental polyads.
Craig Knecht, Example for the sequence.
Craig Knecht, Hemisphinx infinite polyad series.
Craig Knecht, Infinite polyad series construction ideas.
Craig Knecht, Infinite sphinx fundamental polyad series.
Craig Knecht, Insert tiles in T12.
Craig Knecht, Mapping inserts and polyads in frames with 144 tilings.
Craig Knecht, Order 8 fundamental polyads.
Craig Knecht, Order 12 polyad.
Craig Knecht, Polyad overlap.

Cf. A279887.

Conjecture: a(3*k + 2) = 144.

Conjecture: a(3*k + 1) = 180*k + 216.

a(12)-a(20) from Walter Trump, Oct 20 2023

cp's OEIS Frontend

A291582 Maximum number of 6 sphinx tile shapes in a sphinx tiled hexagon of order n.

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A287999 Number of tilings of a stackable 8 sphinx tile shape.

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A317541 Number of tilings of a sphinx of order n with n^2 - 2 elementary sphinxes and a single sphinx domino that has two different tilings.

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A318897 Number of tilings of the even-order sphinx with the two dominoes that form the second-order sphinx.

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A289941 Number of tilings of n interlocking sphinx tiled shape subunits.

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A318778 Number of different positions that an elementary sphinx can occupy in a sphinx of order n.

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A366704 Number of sphinx tilings of T(n+12) with a central T(n) defect where T(k) is an equilateral triangle with side length k.

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