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How many ways are there to tile a rectangle of size $m\times n$ with two types of isosceles triangle, type 1 having area $\frac{1}{2}$ and type 2 having area 1?

I know with only type 1 there are $2^{mn}$ ways, but I don't have any idea how to continue.

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Let an be the number of tilings of an n×1 rectangle. Let bn be the number of tilings of an n×1 rectangle missing a type 1 triangle from one corner.

Then it is clear that an=2bn and bn+1=an+bn, which leads to an+1=3an, and an=2×3n.

There will be a more complicated recurrence for any fixed width of rectangle, but i doubt there is a nice formula for m×n rectangles in general.

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