Recurring Folds Through A Circle

Question

If I were to cut a circular disk of paper, I would have a disk with one section. If I were then to fold it through its centre, I would have a disk with two sections (divided by the crease). If I were then to fold it again through its centre but in a different place to the original folds, I would have a disk with four sections (divided by two creases) etc -> 6, 8, 10, 12 sections.

So if the first zero case is excluded, the pattern is simple and is something along the lines of;

$$ F_n = F_{n-1} + 2, n > 1 $$ $$ F_1 = 2 $$

Which would have the closed solution;

$$ F_n = n \cdot 2, n > 1 $$

My question is, does having no folds rightfully belong in the sequence? It seems to disrupt an otherwise simple relationship.

(Number of folds on the x, regions on the disk on y)

I have a nagging feeling that disregarding what seems to be the first in a sequence feels like cheating. I have been reading 'Concrete Mathematics' after taking a level maths a few years back, and am amazed how different this kind of maths is from school maths. I suppose what I am struggling for is a logical anchor point from which to answer the following:

Is it my mistake to try and include this first point in the sequence, or does it belong and I am just unable to see the recurrence which includes it?

Really appreciate any guidance, Thanks

Answer 1

There is no problem on defining the first member of the sequence independently. Let me give you two simple examples:

• The sequence given by the factorials of the integers, $a_n=n!$ for $n\ge0$, seems to be a closed form which is equal for all $n\ge0$. But, what does $n!$ mean? $$n!= \begin{cases} 1,&n=0,\\[5pt] \prod_{k=1}^n k,&n\ge1. \end{cases}$$ So the term $a_0$ has a particular definition.
• The (probably) most famous sequence, Fibonnaci sequence, has its first TWO elements defined independently. $$F_0=1,\quad F_1=1,\quad F_{n} = F_{n-1}+F_{n-2}\text{ for }n\ge2.$$

By the way, you can find more information about your sequence in the On-Line Encyclopedia of Integer Sequences.