1 . Let $n\ge 1$ be an integer and let $S_n$ be the number of ways in which $n$ can be written as a sum of $1$'s and $3$'s, in which the order of the terms matters. For example, $S_4=3$ because
\[ 4 = 1+1+1+1 = 1+3 = 3+1 \enspace . \]
Which of the following is true, for any integer $n\ge 4$:
(a)
$S_n=S_{n-1}+S_{n-3}$
(b)
$S_n=S_{n-1}+S_{n-2}+S_{n-3}$
(c)
$S_n=3S_{n-3}$
(d)
$S_n=S_{n-1}+S_{n-2}$
(e)
$S_n=3S_{n-1}$