Consider the beginning of the following sequence:1, 11, 21, 1211, 111221, 312211, 13112221, ... .Could you discover the following term and the formation rule of the sequence?

Before you proceed reading, you must stop reading and also think around this difficulty for a moment.

You are watching: 1, 11, 21, 1211, 111221

Let us start with the very first term:

"1", What do you see? The answer is "one 1", you have the right to put it in different way as "11".

So the next term the the succession is "11". What carry out you check out now? "two 1s" the is "21".This term is check out as "one 2 and one 1", the is "1211", the fourth term that the sequence.

Now "1211" is check out as "one 1 one 2 and two 1s", the is "111221". If you read this term is "three 1, 2 2s and one 1", or "312211", the 6th term that the sequence.

Have you master the idea that the formation dominion of the sequence? If so you have the right to say the eight hatchet of the sequence is "1113213211".

For this factor this succession is well-known as look and say sequence and also it was introduced by the brothers mathematician John Conway in 1986.

Roots of the Conway polynomial, plotted through Mathematica.

You might think that this is a mathematical joke. Well, that started like a joke, but we deserve to say some exciting things about this sequence from the mathematical allude of view. We can ask different questions, amongst them:

Besides the numbers 1, 2 and also 3, Do other digits show up in the sequence?What is the growth of size of the basic term of the sequence? This means the following, consider the n-th term of this sequence and also denote through Dn the number of digits that has. For example: D1=1, D2=2, D3=2, D4=4, D5=6 and also so on.

We would prefer to understand the growth rate of Dn, i.e.


The price of the first question is negative. It have the right to be proved that as well as the digits: 1, 2 and 3. No various other digits appear in the regards to the sequence.


The root of this polynomial deserve to be viewed in the figure as blue dots.

A much more sophisticated analysis of this sequence deserve to be done.

References: H. Conway, The weird and wonderful chemistry of audioactive decay, Eureka 46 (1986) 5-16.

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The photo of the root of the polynomial to be done by myself utilizing Mathematica and the polynomial formula v LaTex.