Solution for the sequence of the number of "major flaps" in origami bases and its relation to other sequences

Question

I have recently been getting into origami and reading Robert J. Lang's (a physicist and one of the leading modern origami artist) books. In the book Origami Design Secrets he showed a sequence of more and more complicated origami bases (the starting points for many origami creations).

Here are the first 5 elements of sequence

The basic unit in the base is the right isosceles triangle with two perpendicular folds in it, and you can see it tiled across the bases appearing $2^n$ times in the $n$-th element of the sequence.

Loosely speaking, the $n$-th base is created by tiling the $(n-2)$-th base 4 times in a square pattern. Lang claimed a correspondence between the circular arcs that can be drawn that span the previously described right isosceles triangles (shown in the image) and the number of major "flaps" to use in an origami creation. Each circle, partial or otherwise, corresponds to a flap.

The pattern goes 1, 2, 4, 5, 9, 13, 25 for the first few. I looked up this sequence on OEIS and found nothing, which was a novel experience for me. Does anybody have any ideas how to generalize this sequence and find a closed form solution? Does anybody see any relations to other sequences in geometry? Any help would be greatly appreciated!

Answer 1

What's up, Subhasish! (I'm actually the friend Subhasish mentioned to @AmirSagiv)

To start off with, the first couple terms of the sequence I found with my solution were: $1, 2, 4, 5, 9, 13, 25, 41, 81, 145, 289$.

I found a solution that starts working from the third term. However, it uses a different function for odds and evens.

The first two terms are $1$ and $2$.

Afterwords, odd-indices can be solved with $f(n) = 2^{n-3} + 2^{\frac{n-1}{2}} + 1 = (2^{\frac{n-3}{2}} + 1)^2$

and even-indices with $f(n) = 2^{n-3} + 2^{\frac{n-2}{2}} + 1 = (2^{\frac{n-4}{2}} + 1)^2 + 2^{n-4}$

As @SubhasishMukherjee pointed out in a comment, both of these functions can be consolidated into one form using the ceiling function: $f(n) = 2^{n−3} + 2^{\lceil{\frac{n}{2}}\rceil - 1}+1$.

However, the two separated out formulas are meaningful in their own right. Comparing the sequence and the two functions, you can notice some interesting patterns. Every odd-index term is a square, and the difference between an odd index term and the subsequent even-index term is a power of 4. More specifically, starting from the $n=3$ and $n=5$ pair, the powers of 4 are consecutive powers of 4. Both of these patterns have geometric interpretations that can be noticed if you draw out a few more terms of the sequence.

My solution was a little long, and to be honest, writing math out takes me forever. So I will talk in person with Subhasish (after he wakes up) and some other friends. If we can confirm or disprove my idea then we will edit what I wrote so far.

If anybody online can help to check my work that would be much appreciated as well!