Originally Posted by

**Soroban** Hello, cedricc!

There is nothing wrong with Listing the cases and looking for a pattern.

Quite often, it is the only approach available to us.

I found this list . . .

. . $\displaystyle \begin{array}{cc}

\text{Size} & \text{Tilings} \\ \hline

2\times 1 & 1 \\

2\times 2 & 3 \\

2\times 3 & 5 \\

2\times 4 & 9 \\

2 \times 5 & 15 \\

2\times 6 & 25 \\

2\times 7 & 41 \\

\vdots & \vdots \end{array}$

I see this pattern: Starting with the third term,

. . each number is the sum of the preceding two numbers, plus 1.

That is: .$\displaystyle a_n \;=\;a_{n-1} + a_{n-2} + 1$

That's as far as I dare to go . . .

.