The proof

We want to show that Mn = Mn-2 + Mn-3 + Mn-4
We know that Mn = An + Bn + Cn, therefore we have

Mn-2 + Mn-3 + Mn-4

1. Substitute Mn = An + Bn + Cn for all Mn
= [An-2 + Bn-2 + Cn-2] + [An-3 + Bn-3 + Cn-3] + [An-4 + Bn-4 + Cn-4]

2. Substitute An = Bn+1 for all An
=Bn-1 +Bn-2 + Cn-2 + Bn-2 + Bn-3 + Cn-3 + Bn-3 + Bn-4 + Cn-4

3. Substitute Cn = Bn-1 + Cn-1 for all Bn + Cn
=Bn-1 + Cn-1 + Bn-2 + Cn-2 + Bn-3 + Cn-3

4. Substitute Cn = Bn-1 + Cn-1 for all Bn + Cn
= Cn + Cn-1 + Cn-2

5. Substitute An = Cn-1 for Cn-1 and Cn-2
=Cn + An + An-1

6. Substitute Bn = An-1
=Cn + An + Bn

7. Substitute Mn = An + Bn + Cn
=Mn

Therefore, Mn = Mn-2 + Mn-3 + Mn-4

Finding a pattern

The sequence

1, 1, 1, 2, 3, 4, 6, 9, 13, 19, 28, 41, ...

The pattern
The sum of the red numbers equals the blue number in the following table

It appears that Mn = Mn-2 + Mn-3 + Mn-4, but is it true?

Let's take a look at what we already know. The population in every month is composed of the following age groups: the newborns, the young, and the adults. (The young are simply the rabbits which are no longer newborns, but not yet mature.) Let An represent the number of newborns in the nth month, let Bn represent the number of young in the nth month, and let Cn represent the number of adults in the nth month.

An = Cn-1, because the number of newborns in this month depends on the number of adults in the previous month.
Bn = An-1, because the number of young in this month depends on the number of newborns in the previous month.
Cn = Cn-1 + Bn-1, because the number of adults in this month depends on the number of adults and the number of young in the previous month.

Changing a different condition

Recall

The conditions of the population model which gives us the Fibonacci sequence are
(1) You begin with one pair* of rabbits which have just been born.
(2) Each rabbit reaches sexual maturity after one month.
(3) The gestation period of a rabbit is one month.
(4) A female rabbit will always give birth to one pair of rabbits.
(5) The life span of a rabbit is infinite.

The Project
Previously, I have made changes to the life-span of the rabbits (condition 5). Now, I will make changes to the period of maturity (condition 2). I will first extend the period of maturity from one-month to two-months.

From the figure, we can see that the new sequence is
1, 1, 1, 2, 3, 4, 6, 9, 13, 19, 28, 41, ...

* A pair of rabbits will always imply that one is male and one is female.