Okay, first, let's talk about algebraic degrees and polynomial.

A polynomial can be represented by constants or variables that form them, and they just need to have an end point. These are all polynomials:

1

x

x

^{2}+y

^{3}

The first two are monomials, that is, polynomials of one term.

The degree of a (nonzero) monomial is the sum of the exponents of its variables (all monomials are degree 0 because x

^{0}=1)

The degree of a polynomial is the maximum of the degrees of its terms after the polynomial has been expanded. For example, the polynomial (1+ square root of 2)x

^{2}y

^{3}+12x

^{2}y

^{2}has degree 5.

And that's all we need.

Let's look at the sequence:

-6 (degree 0)

3x-6 (degree 1)

-6x

^{2}+3x-6 (degree 2)

+12x

^{3}-6x

^{2}+3x-6 (degree 3)

...

Once you expand it, you get to this monster:

x

^{71}-x

^{69}-2x

^{68}-x

^{67}+2x

^{66}+2x

^{65}+x

^{64}-x

^{63}-x

^{62}-x

^{61}-x

^{60}-x

^{59}+2x

^{58}+5x

^{57}

+3x

^{56}-2x

^{55}-10x

^{54}-3x

^{53}-2x

^{52}+6x

^{51}+6x

^{50}+x

^{49}+9x

^{48}-3x

^{47}-7x

^{46}-8x

^{45}

-8x

^{44}+10x

^{43}+6x

^{42}+8x

^{41}-5x

^{40}-12x

^{39}+7x

^{38}-7x

^{37}+7x

^{36}+x

^{35}-3x

^{34}+10x

^{33}

+x

^{32}-6x

^{31}-2x

^{30}-10x

^{29}-3x

^{28}+2x

^{27}+9x

^{26}-3x

^{25}+14x

^{24}-8x

^{23}-7x

^{21}+9x

^{20}

+3x

^{19}-4x

^{18}-10x

^{17}-7x

^{16}+12x

^{15}+7x

^{14}+2x

^{13}-12x

^{12}-4x

^{11}-2x

^{10}+5x

^{9}+x

^{7}

-7x

^{6}+7x

^{5}-4x

^{4}+12x

^{3}-6x

^{2}+3x-6

Which is degree 71 and gives this number its connotation.

The result is Conway's Constant:

λ≈1.303577296

But, from where do these numbers come from?

From the Look and Say Sequence.

The look an say sequence is an infinitely growing sequence where, to define the next number, you look and say what you saw in the previous member.

Starting with:

1

What do you see?

A one.

Or, one one. Thus, the next member is:

11.

What do you see?

Two ones.

So the next member is 21.

What do you see?

One two and one one.

So the next member is 1211.

And it goes 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211...

And it never has a number higher than 3.

When you start with number d, where d is an integer different from 1, you get:

d, 1d, 111d, 311d, 13211d, 111312211d, 31131122211d...

As the sequence.

All starting values will keep growing infinitely except for the degenerate (lol, they do have a term for this!) sequence 22, 22, 22...

Anyway, the terms eventually grow in length by about 30% per generation, or, 30.3577296...% per generation (multiply by λ≈1.303577296...)

In other words, if L

_{n}is the number of digits of the nth member of the sequence, then the limit of the ratio L

_{n}+1/L

_{n}(i.e. how bigger the next number of the sequence can be) is:

Conway's Constant (λ≈1.303577296...)

As n approaches infinity.