A360065 - OEIS

%I #16 Oct 02 2024 10:52:56

%S 1,2,45,412,4705,50374,549109,5955544,64683649,702259786,7625147293,

%T 82791470836,898931464993,9760376329678,105975828745957,

%U 1150659965697328,12493588746237697,135652375422278290,1472880803124594061,15992184812239930060,173639288800074705121

%N Number of 3-dimensional tilings of a 2 X 2 X n box using 2 X 1 X 1 dominos and trominos (L-shaped connection of 3 cubes).

%C Recurrence 1 is derived in A359884, "3d-tilings of a 2 X 2 X n box" as a special case of a more general tiling problem: III, example 9.

%H Paolo Xausa, <a href="/A360065/b360065.txt">Table of n, a(n) for n = 0..950</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (7,42,6,-81,27).

%F G.f.: (1 - 5*x - 11*x^2 + 7*x^3) / (1 - 7*x - 42*x^2 - 6*x^3 + 81*x^4 - 27*x^5).

%F Recurrence 1:

%F a(n) = 2*a(n-1) + b(n-1) + c(n-1) + 13*a(n-2) + 2*b(n-2) + c(n-2) + 2*d(n-2),

%F b(n) = 12*a(n-1) + 2*b(n-1) + 2*c(n-1) + e(n-1),

%F c(n) = 16*a(n-1) + 6*b(n-1) + c(n-1) + 2*e(n-1),

%F d(n) = 4*a(n-1) + 2*b(n-1) + d(n-1),

%F e(n) = 16*a(n-1) + 5*b(n-1) + 2*c(n-1) + 2*d(n-1),

%F with a(n), b(n), c(n), d(n), e(n) = 0 for n <= 0 except for a(0)=1.

%F Recurrence 2:

%F a(n) = 7*a(n-1) + 42*a(n-2) + 6*a(n-3) - 81*a(n-4) + 27*a(n-5) for n >= 5.

%F   For n < 5, recurrence 1 can be used.

%e a(2)=45

%e 1) Two parallel trominos and one domino: There are 3 middle axes of the 2 X 2 cube with 4 rotation images each: 12 images.

%e        ___             ___         ___ ___

%e      /__ /|          /   /|      /__ /   /|

%e    /__ /| |___     /__ /  |    /__ /__ /  |

%e   |   | |/__ /|   |   |  /    |   |   |  /|

%e   |   |/__ /| | + |___|/   =  |   |___|/| |

%e   |       | |/                |       | |/

%e   |_______|/                  |_______|/

%e 2) Two "linked" trominos and one domino: 12 rotation images and, as there is no symmetry plane, 12 mirror images: 24 images.

%e        ___                       ___         ___ ___

%e      /   /|                    /   /|      /   /   /|

%e    /__ /  |      _______     /__ /  |    /__ /__ /  |

%e   |   |  /     /__     /|   |   |  /    |   |   |  /|

%e   |   | |  +  |  /__ /  | + |___|/   =  |   |___|/  |

%e   |   | |     |_|   |  /                |   |   |  /

%e   |___|/        |___|/                  |___|___|/

%e 3) Using only dominos: A006253(2)=9 ways, Sum: a(2) = 12 + 24 + 9 = 45.

%t LinearRecurrence[{7, 42, 6, -81, 27}, {1, 2, 45, 412, 4705}, 25] (* _Paolo Xausa_, Oct 02 2024 *)

%Y Cf. A006253, A001045, A033516, A335559, A359884, A359885, A360064, A360066.

%K nonn,easy

%O 0,2

%A _Gerhard Kirchner_, Jan 30 2023