What’s the pattern of the following sequence?

Can you guess the next term?

The answer is 60. (Honestly, I don’t think I can if I haven’t worked on this problem.) The pattern is:

This sequence looks artificial. But I will show that it has an interesting “geometric” property. The sequence is very similar to the Fibonacci sequence . Recall that is defined by

So the first few terms of are:

Recently I learned the following amazing property of (while teaching calculus):

Theorem 1

For example,

The proof of this result can be found in this video. In fact the identity is proved by computing the areas of the “Fibonacci rectangles” in two ways.

Inspired by this result, I tried to look for a sequence with a similar property. After playing with sequences for a while, I found:

Theorem 2For the sequence defined in (1),

(Unfortunately, it is not as beautiful as Theorem 1.) For example,

**Proof**: For a rectangular block of width , depth and height , we record this information as

*
*

here stands for dimension. (Of course, its volume is also .)

In the following figure, there is a single cube of dimension

Now, we “thicken” this cube in the positive -direction by units. This just means appending a (red) cube of dimension to the right hand side of the original cube (assuming that the positive direction is pointing to the right):

So now the new rectangular block is of dimension

by the definition of . Let’s denote this “thickening” (or appending) process by ““.

Now, we do a “” process to . More precisely, we thicken this block () in the positive -direction by units. Equivalently, this amounts to appending a (red) rectangular block of dimension just beside in the positive -direction:

Let’s call the resulting block by . Then by the definition of ,

To go on, we do a process to to obtain , which amounts to appending a block of dimension to (in the positive -direction). Note that in each process, after appending a new block to , the resulting block is still a rectangular blocks, since the dimension fits each other.

We recursively do these processes to obtain . i.e. is obtained from by doing a “” to , etc. In short:

It’s now easy to verify our claim. For simplicity, let’s look at . On the one hand, its volume is clearly

On the other hand, since is obtained by successively appending rectangular blocks starting from , it is easy to see that

Therefore

* The general case can be similarly proven. *

**Question: **Can you generalize this to “higher dimensions”?