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

%I A291582 #108 Jan 17 2025 09:42:51
%S A291582 30,132,306,552,870,1260,1722,2256,2862,3540,4290,5112,6006,6972,8010,
%T A291582 9120,10302,11556,12882,14280,15750,17292,18906,20592,22350,24180,
%U A291582 26082,28056,30102,32220,34410,36672,39006,41412,43890,46440,49062,51756,54522,57360,60270,63252
%N A291582 Maximum number of 6 sphinx tile shapes in a sphinx tiled hexagon of order n.
%C A291582 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 A291582 _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 A291582 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 A291582 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 A291582 G. C. Greubel, <a href="/A291582/b291582.txt">Table of n, a(n) for n = 1..5000</a>
%H A291582 Craig Knecht, <a href="/A291582/a291582_4.png">Eight sphinx tile frame tessellations.</a>
%H A291582 Craig Knecht, <a href="/A291582/a291582_1.png">Example for the sequence</a>.
%H A291582 Craig Knecht, <a href="/A291582/a291582_2.png">Order 4 Hexagon with 246 subshapes</a>.
%H A291582 Craig Knecht, <a href="/A291582/a291582_3.png">Sphinx Tile Component Triangles</a>.
%H A291582 Craig Knecht, <a href="/A291582/a291582.png">Sphinx tiling of the triangle used to make the hexagon</a>.
%H A291582 Craig Knecht, <a href="/A291582/a291582_5.png">T12 symmetric sphinx tilings.</a>
%H A291582 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 A291582 Wikipedia, <a href="https://en.wikipedia.org/wiki/Walter_Trump">Walter Trump</a>.
%H A291582 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A291582 a(n) = 6*n*(6*n-1). - _Walter Trump_
%F A291582 G.f.: 2*x*(15+21*x)/(1-x)^3. - _Vincenzo Librandi_, Sep 20 2017
%F A291582 a(n) = 6*A049452(n) = 6*n*A016969(n-1). - _Torlach Rush_, Nov 28 2018
%F A291582 E.g.f.: 6*exp(x)*(5 + 17*x + 6*x^2). - _Stefano Spezia_, Dec 07 2018
%F A291582 a(n) = A016970(n-1) + A016969(n-1). - _Torlach Rush_, Dec 10 2018
%F A291582 From _Amiram Eldar_, Jul 30 2024: (Start)
%F A291582 Sum_{n>=1} 1/a(n) = log(2)/3 + log(3)/4 - sqrt(3)*Pi/12.
%F A291582 Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/6 - log(2)/6 - arccoth(sqrt(3))/sqrt(3). (End)
%p A291582 seq(6*n*(6*n-1),n=1..100); # _Robert Israel_, Sep 19 2017
%t A291582 Array[6 # (6 # - 1) &, 42] (* _Michael De Vlieger_, Sep 19 2017 *)
%t A291582 CoefficientList[Series[2(15 + 21 x)/(1-x)^3,{x, 0, 50}], x] (* _Vincenzo Librandi_, Sep 20 2017 *)
%t A291582 CoefficientList[Series[6 E^x (5 + 17 x + 6 x^2), {x, 0, 50}], x]*
%t A291582 Table[n!, {n, 0, 50}] (* _Stefano Spezia_, Dec 07 2018 *)
%o A291582 (Magma) [6*n*(6*n-1): n in [1..50]]; // _Vincenzo Librandi_, Sep 20 2017
%o A291582 (PARI) a(n) = 6*n*(6*n-1); \\ _Altug Alkan_, Apr 08 2018
%o A291582 (Sage) [6*n*(6*n-1) for n in (1..50)] # _G. C. Greubel_, Dec 04 2018
%o A291582 (GAP) List([1..30], n -> 6*n*(6*n-1)); # _G. C. Greubel_, Dec 04 2018
%Y A291582 Cf. A016969, A049452.
%Y A291582 Cf. A279887, A287999.
%K A291582 nonn,easy
%O A291582 1,1
%A A291582 _Craig Knecht_, Aug 30 2017