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A386222 Number of 3-dimensional tilings of a 2 X 2 X (n+1) box with the two upper right cells removed, using 2 X 2 X 1 plates and 1 X 2 X 1 dominos.

%I A386222 #21 Aug 26 2025 06:31:46
%S A386222 1,5,34,201,1241,7538,46045,280693,1712338,10443297,63697825,
%T A386222 388506066,2369604597,14452808029,88151396594,537657790873,
%U A386222 3279312211305,20001361622066,121993408939853,744068928339589,4538266259447698,27680043927136849,168827650973959281
%N A386222 Number of 3-dimensional tilings of a 2 X 2 X (n+1) box with the two upper right cells removed, using 2 X 2 X 1 plates and 1 X 2 X 1 dominos.
%C A386222 Here is the box for n=3:
%C A386222     ____________
%C A386222    /   /   /   /|
%C A386222   /___/___/___/ |____
%C A386222  /   /   /   /| /   /|
%C A386222 /___/___/___/ |/___/ |
%C A386222 |   |   |   | /   /| /
%C A386222 |___|___|___|/___/ |/
%C A386222 |   |   |   |   |  /
%C A386222 |___|___|___|___| /.
%H A386222 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,9,-14).
%F A386222 G.f.: 1/(1 - 5*x - 9*x^2 + 14*x^3).
%F A386222 a(n) = 5*a(n-1) + 9*a(n-2) - 14*a(n-3) for n >= 3.
%F A386222 a(n) = A359884(n) + 2*a(n-1).
%e A386222 Here is one of the a(1)=5 ways to tile the shape for n=1, in this case with one flat plate on the bottom and one domino on top.
%e A386222     ____
%e A386222    /   /|
%e A386222   /   / |____
%e A386222  /   /  /   /|
%e A386222 /___/  /   / |
%e A386222 |   | /   /  /
%e A386222 |___|/___/  /
%e A386222 |       |  /
%e A386222 |_______| /.
%t A386222 LinearRecurrence[{5, 9, -14}, {1, 5, 34}, 30]
%Y A386222 Cf. A359884
%K A386222 nonn,changed
%O A386222 0,2
%A A386222 _Greg Dresden_ and Xiaoya Gao, Aug 13 2025