Find sequence of specific binary combination

Question

We have a set $S_n$ containing $n$ binary values, lets say $S_6$ = [0,0,1,1,0,1].

I want to make sure that only one consecutive 0 may occur between the 1's.

$P_6$ = [1,0,1,1,0,1] Allowed.

$Q_6$ = [1,1,0,0,0,0] Allowed.

$U_6$ = [1,1,0,0,0,1] Not allowed.

The problem can be divided into two parts:

1. Look for occurrence of sequence [1,0,0].
2. If found, make sure that the sum of all the consecutive elements adds up to 0.

How do you express this in mathematical terms?