11/12/23 and the Slow Fibonacci Sequence

Today is 11/12/23, which feels like it’s just short of being an important mathematical date. Instead of working on something for NaNoWriMo, I was staring at the date, trying to find an interesting pattern in the digits 1,1,1,2,2,3… and I think I’ve stumbled upon one. Presenting: The Slow Fibonacci Sequence!

The regular Fibonacci sequence is ubiquitous: add up the two latest numbers in the sequence, and you get the next one. It starts with 1, 1, and anyone can compute as many terms as they want by just adding the last two numbers.

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610…

And the recursive formula for the sequence can be elegantly expressed like this:

F_n = F_n-1 + F_n-2

Now, the date today doesn’t fit with this – it would have to be November 23, 2058 for us to get a real Fibonacci date – but it somehow perfectly fits with a degenerate form of the Fibonacci sequence. Instead of adding up the latest two terms to get the next one, skip a term and add up the two previous terms of the sequence to get the next one. The explanation’s ugly in English, but check out this beautiful recursive formula, almost exactly like the last one:

 F_n = F_n-2 + F_n-3

So, what does this sequence look like? Instead of starting with two numbers, we need to start with three. (Otherwise, we’d be able to figure out the fourth term of the sequence, but not the third or any others!) The Fibonacci sequence starts with two ones, so the logical start is three ones:

1, 1, 1

Now, we skip the last term and add the previous two:

1, 1, 1, 2

Then we do it again, skipping the new 2 to add the two middle terms:

1, 1, 1, 2, 2

And then we iterate again:

1, 1, 1, 2, 2, 3

And repeating this pattern, we can expand out this sequence:

1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65…

As you can see, it’s growing, but not nearly as fast as the ordinary Fibonacci sequence. And our date, 11/12/23, fits with it perfectly!

Now, of course, we have to ask – is it interesting? I’m going to play with the numbers a bit, and rate the things I find from interesting to not interesting.

Mildly Interesting – Substituting variables for the first three terms of the sequence gives us a bit of a better look in terms of other versions of the sequence that don’t start with 1, 1, 1. The first few terms of the sequence can be written as:

a, b, c, a + b, b + c, a + b + c, a + 2b + c, a + 2b + 2c,  2a + 3b + 2c, 2a + 4b + 3c, 3a + 5b + 4c, 4a + 7b + 5c…

 

Mildly Interesting – The formulaic view above gives us insight into what the next term will be for all terms in the sequence, not just based on the first few terms. If you knew what the 100^th, 101^st, and 102^nd terms of the sequence were, and you needed the 110^th term, you wouldn’t have to backtrack, or try and figure out what the formula for the 110^th would be – the formula for the term eight places ahead of your three known terms is 3a + 5b + 4c, so you could just plug in the three terms you had for a, b, and c, and get your answer. 

 

Barely Interesting - We do have these formulas for every term, so the sixth term isn't inherently more interesting than the fifth or seventh terms, but it's interesting because it's a + b + c... which means that another way to define this sequence is:

F_n = F_n-3 + F_n-4 + F_n-5

Of course, the latter two add up to F_n-2 anyways, so it’s not that huge. Also, it’s not as tight of a definition as the first one – there are a lot of other sequences that fulfill this definition but not the others.

 

Quite Interesting – Due to the way that the sequence works, we can look at the individual coefficients of each variable in the formulaic expansion above… and they’re following the same pattern as the sequence itself, and following it slightly out of phase!

Starting at term 6, the a’s are a, a, a, 2a, 2a, 3a, 4a… and the c’s follow the same pattern, but start at term 5, and the b’s start at term 4! This sync disparity is what makes the three little mini sequences add up to an identical, but several-terms-ahead version of themselves!

Interestingly, the ordinary Fibonacci sequence does this too! If we do the same variable substitutions for it, we get:

a, b, a+b, a+2b, 2a + 3b, 3a + 5b, 5a + 8b, 8a + 13b, 13a + 21b, 21a + 34b…

And there they are, the coefficients are the Fibonacci sequence themselves – the b’s start on term 2, and the a’s start on term 3!

 

We haven’t even dug into other degenerate Fibonacci sequences with all sorts of rules, but that’ll have to wait, because I need to get this out today for it to be relevant. Happy Slow Fibonacci Day, everyone!