Colouring a Strip
Problem 670
A certain type of tile comes in three different sizes - 1×1, 1×2, and 1×3 - and in four different colours: blue, green, red and yellow. There is an unlimited number of tiles available in each combination of size and colour.
These are used to tile a rectangle, where is a positive integer, subject to the following conditions:
For example, the following is an acceptable tiling of a rectangle:
but the following is not an acceptable tiling, because it violates the "no four corners meeting at a point" rule:
Let be the number of ways the rectangle can be tiled subject to these rules. Where reflecting horizontally or vertically would give a different tiling, these tilings are to be counted separately.
For example, , , and .
Find .