Valentin Vornicu wrote: How many ways are there to divide a $2\times n$ rectangle into rectangles having integral sides, where $n$ is a positive integer? This seems interesting, however, are two divisions considered different if they differ only in the placement of the different subrectangles? Daniel

to dhernandez: I think it should be considered different. and based on this ,I got the number is $\frac{4+\sqrt{2}}{14}*(3+\sqrt{2})^n+\frac{4-\sqrt{2}}{14}*(3-\sqrt{2})^n$

Hawk Tiger could you please post your solution

You can use the fact that every rectangle placed across(parallel to the side of length $2$) will create two $2Xk$ rectangles, and then you could find a recursion relation.

Well i solved this problem, but i took a lot of time calculating the final formula. I found the recurrence $r_{n}= 6r_{n-1}-7 r_{n-2}$ this was easy with $r_{1}= 2$ and $r_{2}=8$. I know how to solve this recurrence relations in two ways, gerenating functions or solving the caracteristic equation. Is there a faster way of finding the general term? I don't think there exists a way to count this without using recurrences, is there? I hope someone can help.

jasamper88 wrote: Well i solved this problem, but i took a lot of time calculating the final formula. I found the recurrence $ r_{n} = 6r_{n - 1} - 7 r_{n - 2}$ this was easy with $ r_{1} = 2$ and $ r_{2} = 8$. I know how to solve this recurrence relations in two ways, gerenating functions or solving the caracteristic equation. Is there a faster way of finding the general term? I don't think there exists a way to count this without using recurrences, is there? I hope someone can help. Please explain how did you get that recursion.