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.
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:
a) A0->3*A1 (3 ways of selecting a pair of p4-plates, also
A001045(2)=3).
b) A0->9*A2 (9 ways of tiling a 2 X 2 X 2 cube with 3d-dominos, also
A006253(2)=9).
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.
d) B0->1*A1 (one accomplishing p4-plate is placed on B0).
e) B0->*2B1 (2 ways of selecting a pair of dominos on B0).
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.
/\ /\ /\
/ \ / \ / \
/ \ S' /\ / \ /\ / \ /\
/ \ / \ / \ / \ / \ / \
|\ / \ /||\ / \ /||\ / \ /|
| \ / \ / || \ / \ / || \ / \ / |
| S |\ /| R'|| |\ /| R"|| |\ /| |---> 4th dimension
|\ | \ / | /||\ | \ / | /||\ | \ / | /|
| \| R | |/ || \| | |/ || \| | |/ |
| P |\ | /| Q'|| |\ | /| Q"|| |\ | /| |
\ | \|/ | / \ | \|/ | / \ | \|/ | /
\| Q | |/ \| | |/ \| | |/
\ | / \ | / \ | /
\|/ \|/ \|/
