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

Author

Craig Knecht, Aug 30 2017

Keywords

• nonn
• easy

Comments

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

Links

• 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).

Crossrefs

Cf. A016969, A049452.

Cf. A279887, A287999.

Programs

GAP

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

Magma

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

Maple

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

Mathematica

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 *)

PARI

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

Sage

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

Formula

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)