Proof of the second formula

The proof of the second formula is carried out similarly to the proof of the first formula.

Sn = Sn-2 + Sn-3 + Sn-4 (second formula)

Let An represent the number of newborns in the nth month
Let Bn represent the number of 1-month olds in the nth month
Let Cn represent the number of 2-month olds in the nth month, and
Let Dn represent the number of 3-month olds in the nth month

Because the population in each month only consists of these age groups, we can say that Sn = An + Bn + Cn + Dn. (1)

Also, because the number of newborns depends on the number of adults (mature rabbits) in the previous month, we can say that
An = Bn-1 + Cn-1 + Dn-1. (2)

Also, because the rabbits age, we have
Bn = An-1 (3)

Cn = Bn-1 (4)

Dn = Cn-1 (5)

Now we can use formulas 1-5 to prove (or disprove) the second formula.

First, substitute (1) into each term in the second forumla:

Sn = An + Bn + Cn + Dn
Sn-2 = An-2 + Bn-2 + Cn-2 + Dn-2
Sn-3 = An-3 + Bn-3 + Cn-3 + Dn-3
Sn-4 = An-4 + Bn-4 + Cn-4 + Dn-4

Set up the LHS and RHS:

LHS: An + Bn + Cn + Dn =
RHS: An-2 + Bn-2 + Cn-2 + Dn-2 + An-3 + Bn-3 + Cn-3 + Dn-3 + An-4 + Bn-4 + Cn-4 + Dn-4

Use (2), (3), (4), and (5) to substitute into the RHS until it equals the LHS.

LHS: An + Bn + Cn + Dn =
RHS: An + Bn + Cn + Dn

Therefore, we can generate the 4MLS sequence as follows:
when n is less than or equal to 4: Fn = Fn-1 + Fn-2
when n is greater than 4: Sn = Sn-2 + Sn-3 + Sn-4