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%I #31 Jun 24 2024 15:53:35
%S 1,3,24,133,839,5056,30969,188603,1150952,7018621,42811231,261110416,
%T 1592592465,9713598835,59245780536,361354997685,2203996629559,
%U 13442737199456,81990685695721,500082110459883,3050128402768520,18603511408241453,113467563119685583
%N Number of 3-dimensional tilings of a 2 X 2 X n box using 2 X 2 X 1 plates and 1 X 2 X 1 dominos.
%C The first recurrence is derived in "3d-tilings of a 2 X 2 X n box" as a special case of a more general tiling problem: III, example 4.
%H Paolo Xausa, <a href="/A359884/b359884.txt">Table of n, a(n) for n = 0..1000</a>
%H Gerhard Kirchner, <a href="/A359884/a359884.txt">Maxima code</a>
%H Gerhard Kirchner, <a href="/A359884/a359884_1.pdf">3d-tilings of a 2X2Xn box</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,9,-14).
%F G.f.: (1 - 2*x) / (1 - 5*x - 9*x^2 + 14*x^3).
%F a(n) = 3*a(n-1) + c(n-1) + 7*a(n-2) where c(n) = 8*a(n-1) + 2*c(n-1) with a(n),c(n) <= 0 for n <= 0 except for a(0)=1.
%F a(n) = 5*a(n-1) + 9*a(n-2) - 14*a(n-3) for n >= 3.
%e a(1) = 3
%e _______ _______ _______
%e / /| / / /| /______ /|
%e /______ / | /__ /__ / | /______ /||
%e | | / | | | / | ||/
%e |_______|/ |___|___|/ |_______|/
%t LinearRecurrence[{5, 9, -14}, {1, 3, 24}, 25] (* _Paolo Xausa_, Jun 24 2024 *)
%o (Maxima) /* See link "Maxima code". */
%Y Cf. A006253, A001045, A335559, A359885, A359886.
%K nonn,easy
%O 0,2
%A _Gerhard Kirchner_, Jan 20 2023
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