# A Loopy Sequence For Father’s Day

Happy Father’s Day!

Today’s the day that dads get ties and get to tell jokes with impunity. Or if you’re a mathematician, you get a book on knots so you can tie the perfect tie.

Now, I was able to get a Trinity tie put together.

So my job now is to learn how to tie the Eldredge. It feels like I’m going to need a necktie that’s about 7 feet long to make this work. But hot damn. Source: https://i.imgur.com/8ug05FA.jpg

My wonderful wife got me The 85 Ways to Tie a Tie: The Science and Aesthetics of Tie Knots

Also for today, I’ll put forth a “Father’s Day Sequence”. Keep reading to find out more!

### First, let’s have some shoutouts from the Twitterverse.

We have this unimpeachable gem from @mathtans

Make sure to check out his website at https://mathtans.blogspot.com/.

And we have this from MichLampinen

### Now getting back to ties

Ties are knots. Knots have loops. Let’s talk about a loopy sequence!

Let $$0,4,6,8,9$$ be “loopy digits” (can you guess why?) and $$1,2,3,5,7$$ be “stranded digits”.

Now, let’s make up our loopy sequence — an increasing sequence of non-negative integers with a special construction rule. First, some preliminaries. We will keep track of the number of “loops” for any given term in our loopy sequence. For the loopy digits, 0 has one loop, 4 has one loop, 6 has one loop, 8 has two loops, and 9 has one loop. The stranded digits have zero loops. A number like 63891 has four loops.

Let’s see how we construct our loopy sequence.

Our loopy sequence begins at 0. Thus, $$a_{0} = 0$$ and if $$n_{k}$$ is the number of loops for the $$k$$th term of the sequence, then $$n_{0} = 1$$. The next number in the sequence, $$a_{1}$$ is the first integer greater than $$a_{0}$$ such that $$n_{1} \geq n_{0}$$. Thus, $$a_{1} = 4$$. We should be able to see that $$a_{2} = 6$$. Do you believe that we must have $$a_{3} = 8$$? Since $$n_{2} = 1$$, we want the first integer greater than $$a_{2}$$ such that $$n_{3} \geq n_{2}$$. The first integer that does this is $$a_{3} = 8$$. Take a few minutes to reason this out if it seems confusing. If after a few minutes it is still confusing, then keep reading, maybe more examples will clarify the situation.

Now, what is $$a_{4}$$. It’s not 9 since $$n_{4} = 1$$ and we are already at $$n_{3} = 2$$. So what’s the next integer that has at least two loops? It must be $$a_{4} = 18$$. Makes sense?

Here are the first few terms of our loopy sequence.
$$0, 4, 6, 8, 18, 28, 38, 40, 44, 46, 48, 68, 80, 84, 86, 88$$
and here is what the loop count sequence
$$1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4$$

Here are a few questions for you to work out.

#### If we wanted to create a different loopy sequence, $$B$$, with $$b_{0} > 0$$, then prove or disprove that there exists $$k, j \in \mathbb{N}$$ such $$a_{k + i} = b_{j + i}$$ for all non-negative integer $$i$$.

In other words, how we start the loopy sequence doesn’t matter. Every loopy sequence will eventually align with our original construction. For example, if $$b_{0} = 11$$, then we have $$11, 12, 13, 14, 16, 18, 28, \ldots$$ and we see that $$a_{4 + i} = b_{5 + i}$$ for all $$i \geq 0$$.

### While You’re Here …

Interested in knowing what Primal Words are? How about Funny Numbers? Or Semi r-Primes?

Happy Father’s Day! 