cp's OEIS Frontend

A360645 Number of 4-dimensional tilings of a 2 X 2 X 2 X n box with 2 X 2 X 1 X 1 plates.

%I A360645 #16 May 28 2024 17:56:52
%S A360645 1,3,30,177,1281,8520,58629,397887,2715510,18490533,126023349,
%T A360645 858595560,5850498441,39863005323,271617783150,1850725023657,
%U A360645 12610357769721,85923544106760,585460036653789,3989166905015367,27181111280961990,185204779320272253
%N A360645 Number of 4-dimensional tilings of a 2 X 2 X 2 X n box with 2 X 2 X 1 X 1 plates.
%C A360645 The figure shows three 2 X 2 X 2 cubes as intersections of three successive hyperplanes (distance 1) with the box. The 3-d cross-section of a 2 X 2 X 1 X 1 plate is a 2 X 2 X 1 plate (p4) as part of one cube or a 2 X 1 X 1 domino if the plate (p2) connects two cubes. p4 or p2 indicates the number of unit cubes on the current level (hyperplane). PQRS and P'Q'R'S' (not visible: P') is one of three ways to select a pair of p4-plates. Q'Q"R"S' represents a p2-plate.
%C A360645 Suppose the box is completely tiled up to a certain level. Then the next (current) level may be empty (profile A0) or not (profile B0). The index 0 is used for the current level and continued with 1,2... Transitions:
%C A360645 a) A0->3*A1 (3 ways of selecting a pair of p4-plates, also A001045(2)=3).
%C A360645 b) A0->9*A2 (9 ways of tiling a 2 X 2 X 2 cube with 3d-dominos, also A006253(2)=9).
%C A360645 c) A0->12*B1. One p4-plate and two p2-plates can be selected in 12 ways: 6 faces of the 2 X 2 X 2 cube and two ways of selecting a pair of dominos on each face. They tile the next level with corresponding dominos. A further nonempty profile does not occur. Also, A359884(2)-A006253(2)-A001045(2)=24-9-3=12.
%C A360645 d) B0->1*A1 (one accomplishing p4-plate is placed on B0).
%C A360645 e) B0->*2B1 (2 ways of selecting a pair of dominos on B0).
%C A360645 Let a(n) and b(n) be the number of tilings of the 2 X 2 X 2 X n box ending with an A- or a B-profile respectively. With the transitions above, one obtains recurrence 1.
%C A360645          /\               /\               /\
%C A360645        /    \           /    \           /    \
%C A360645      / \  S' /\       / \     /\       / \     /\
%C A360645    /     \ /    \   /     \ /    \   /     \ /    \
%C A360645   |\     / \     /||\     / \     /||\     / \     /|
%C A360645   |  \ /     \ /  ||  \ /     \ /  ||  \ /     \ /  |
%C A360645   | S |\     /| R'||   |\     /| R"||   |\     /|   |---> 4th dimension
%C A360645   |\  |  \ /  |  /||\  |  \ /  |  /||\  |  \ /  |  /|
%C A360645   |  \| R |   |/  ||  \|   |   |/  ||  \|   |   |/  |
%C A360645   | P |\  |  /| Q'||   |\  |  /| Q"||   |\  |  /|   |
%C A360645    \  |  \|/  |  /  \  |  \|/  |  /  \  |  \|/  |  /
%C A360645      \| Q |   |/      \|   |   |/      \|   |   |/
%C A360645        \  |  /          \  |  /          \  |  /
%C A360645          \|/              \|/              \|/
%H A360645 Paolo Xausa, <a href="/A360645/b360645.txt">Table of n, a(n) for n = 0..1000</a>
%H A360645 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,15,-18).
%F A360645 Recurrence 1: a(n) = 3*a(n-1) + b(n-1) + 9*a(n-2), b(n) = 12*a(n-1) + 2*b(n-1), with a(0) = 1 and a(-1) = b(0) = 0.
%F A360645 Recurrence 2: a(n) = 5*a(n-1) + 15*a(n-2) - 18*a(n-3).
%F A360645 G.f.: (1-2*x) / (1-5*x-15*x^2+18*x^3).
%t A360645 LinearRecurrence[{5, 15, -18}, {1, 3, 30}, 25] (* _Paolo Xausa_, May 28 2024 *)
%Y A360645 Cf. A001045, A006253, A359884.
%K A360645 nonn
%O A360645 0,2
%A A360645 _Gerhard Kirchner_, Feb 15 2023