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

1,1

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, 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.

Carlos Rivera, Puzzle 920. An enigma related to A291582, Primes puzzles and problems connections.

Wikipedia, Eight sphinx tile tessellation of the same hexagon

Wikipedia, Hyper-Packing the Sphinx

Wikipedia, Symmetric sphinx tiled triangles

Wikipedia, Walter Trump

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

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

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 *)

Prog

(Magma) [6*n*(6*n-1): n in [1..50]]; // Vincenzo Librandi, Sep 20 2017
(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
(GAP) List([1..30], n -> 6*n*(6*n-1)); # G. C. Greubel, Dec 04 2018

Crossrefs

Cf. A016969, A049452.

Cf. A279887, A287999.

Sequence in context: A044362 A044743 A221522 * A079588 A100147 A117750

Adjacent sequences: A291579 A291580 A291581 * A291583 A291584 A291585

Keyword

nonn,easy

Author

Craig Knecht, Aug 30 2017

Status

approved