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Just For Fun

The Look-and-Say Sequence

In a previous blog I looked in more detail into “the daddy of all integer sequences” – the Fibonacci sequence. While the Fibonacci sequence is an impossible sequence to match up to, there are plenty of other intriguing integer sequences out there. So if you can deal with the disappointment of them not being as awesome as the Fibonacci sequence I hope you will find them as interesting as me.

The Online Encyclopedia of Integer Sequences

Before we focus on the sequence in question, there is an online goldmine for integer sequences. This is called the Online Encyclopedia of Integer sequences (OEIS). The database was created in 1996 by mathematician Neil Sloane. Sloane is a regular contributor to the excellent Youtube channel Numberphile. Sloane actually started collating interesting integer sequences long before this. He originally published two books of integer sequences – “A Handbook of Integer Sequences” in 1973 which contained 2,372 sequences and an even bigger book in 1995 “The Encyclopedia of Integer Sequences” which contained 5,488 sequences. However, as you can imagine, Sloane soon had too many integer sequences to publish in a book that can be carried under the arm of a human being. The timely invention of the internet led to the online encyclopedia being created.

Obviously there are infinitely many possible integer sequences so the site cannot hold every single integer sequence. Now, anyone can put forward a sequence for the website, but there are a team of volunteers who ensure each sequence is worthy of making the cut and wont lower the high standards of the encyclopedia. Each sequence is given a unique code, has a brief description of how the sequence is generated as well as links to relevant academic papers on the sequence so you can read up even more on each sequence. However by March 2021 there were over 340,000 integer sequences on the website. All the big hitters are there – The Fibonacci Sequence is there of course (A000045) and so is a sequence of all prime numbers (A000040).

Over the coming months and years I will be posting blogs on some of my favourites from the OEIS, explaining how they are generated and why they are interesting. Today’s sequence is the “Look-and-say sequence” (A005150).

The Look-and-Say Sequence

This is very much a Ronseal sequence – “it does exactly what it says on the tin”.

Like the Fibonacci sequence, it is not unique and a different sequence can be generated by picking the two starting numbers. You can generate many different look-and-say sequences by choosing a different starting number, or “seed” as I will refer to them as.

For simplicity let’s begin with the look-and-say sequence that begins with 1.

The next term in the sequence is generating by describing the previous term in the sequence.

The first term has one 1 in it.

So the second term is 11, because the previous term has one 1 (1 1).

The third term is 21, because the previous term has two 1’s (2 1).

The fourth term is 1211, because the previous term has 1 2 and 1 1 (1 2 1 1).

The fifth term is 111221, because the previous term has 1 1, 1 2 and 2 1’s (1 1 1 2 2 1).

The sixth term is 312211, because the previous term has 3 1’s, 2 2’s and 2 1’s (3 1 2 2 2 1).

As you can see, these numbers quickly get very large.

Here are the first 10 terms in the sequence:

1,
11,
21,
1211,
111221,
312211,
13112221,
1113213211,
31131211131221,
13211311123113112211

One interesting property of this sequence is that it only ever contains the numbers 1, 2 and 3.

And a consequence of this property is that no number in the sequence can contain more than 3 of the same digit consecutively. If it did, then the next number in the sequence would contain a 4.

How does the sequence look with other seed numbers?

So we only looked at the most common look-and-say sequence there with a seed (starting number) of 1.

We can create a look-and-say sequence with any seed.

Here are the first few terms of the sequence with seed 2:

2
12
1112
3112
132212
1113221112
3113223112
13211322132112

And here it is with seed 3:

3
13
1113
3113
132113
1113122213
311311321113
1321132113123113

Again you notice that these two sequences never contain any digit higher than 3.

Obviously this rule no longer applies when we use a seed number larger than 3. For example, here is the look-and-say sequence with seed 4:

4
14
1114
3114
132114
1113122114
311311222114
13211321322114

You will notice that the digit 4 only appears in this sequence as the last digit.

You may also notice that in all of these sequences, each number always ends with the seed number.


I hope you found that interesting. For more sequence-related content, check out these blogs:

Different types of sequences in GCSE Maths
How to find the nth term of a linear sequence
How to find the nth term of a quadratic sequence (note this is for higher tier only)
The Fibonacci sequence and the Golden Ratio

You can also try out our nth term worksheets to practice exam-style sequence questions:

Nth term
Quadratic sequences (note this is for higher tier only)
Patterns and sequences

If you are in Bristol and you or a son/daughter needs some extra help with mathematics, book in a free taster session.

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