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 A291582 Maximum number of 6 sphinx tile shapes in a sphinx tiled hexagon of order n. 4
 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 (list; graph; refs; listen; history; text; internal format)
 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 tiling of the triangle used to make the hexagon. Carlos Rivera, Puzzle 920. An enigma related to A291582, Primes puzzles and problems connections. 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 * A100147 A079588 A117750 Adjacent sequences:  A291579 A291580 A291581 * A291583 A291584 A291585 KEYWORD nonn,easy AUTHOR Craig Knecht, Aug 30 2017 STATUS approved

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Last modified October 15 20:04 EDT 2019. Contains 328037 sequences. (Running on oeis4.)