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A291582 Maximum number of 6 sphinx tile shapes in a sphinx tiled hexagon of order n. 4


%S 30,132,306,552,870,1260,1722,2256,2862,3540,4290,5112,6006,6972,8010,

%T 9120,10302,11556,12882,14280,15750,17292,18906,20592,22350,24180,

%U 26082,28056,30102,32220,34410,36672,39006,41412,43890,46440,49062,51756,54522,57360,60270,63252

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

%C 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.

%C Walter Trump enumerated all 830 sphinx tilings of this triangle and found six symmetrical examples one of which is used to produce this sequence.

%C 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.

%C 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

%H Craig Knecht, <a href="/A291582/a291582_1.png">Example for the sequence.</a>

%H Craig Knecht, <a href="/A291582/a291582_2.png">Order 4 Hexagon with 246 subshapes.</a>

%H Craig Knecht, <a href="/A291582/a291582.png">Sphinx tiling of the triangle used to make the hexagon.</a>

%H Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_920.htm">Puzzle 920. An enigma related to A291582</a>, Primes puzzles and problems connections.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/File:Sphinx_Tile_Patterns.png">Eight sphinx tile tessellation of the same hexagon</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/File:Hyper-packing_the_sphinx.png">Hyper-Packing the Sphinx</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/File:Sphinx_tile_triangle.png">Symmetric sphinx tiled triangles</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Walter_Trump">Walter Trump</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F a(n) = 6*n*(6*n-1). - Walter Trump

%F Cf.: 2*x*(15+21*x)/(1-x)^3. - _Vincenzo Librandi_, Sep 20 2017

%p seq(6*n*(6*n-1),n=1..100); # _Robert Israel_, Sep 19 2017

%t Array[6 # (6 # - 1) &, 42] (* _Michael De Vlieger_, Sep 19 2017 *)

%t CoefficientList[Series[2(15 + 21 x)/(1-x)^3,{x, 0, 50}], x] (* _Vincenzo Librandi_, Sep 20 2017 *)

%o (MAGMA) [6*n*(6*n-1): n in [1..50]]; // _Vincenzo Librandi_, Sep 20 2017

%o (PARI) a(n) = 6*n*(6*n-1); \\ _Altug Alkan_, Apr 08 2018

%Y Cf. A279887, A287999.

%K nonn,easy

%O 1,1

%A _Craig Knecht_, Aug 30 2017

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Last modified October 19 07:21 EDT 2018. Contains 316337 sequences. (Running on oeis4.)