Calculate some terms to identify a pattern.

$a_3 = a_{(3-1)} - a_{(3-2)} = a_2 - a_1 = 4 - (-5) = 9$
$a_4 = a_3 - a_2 = 9 - 4 = 5$
$a_5 = a_4 - a_3 = 5 - 9 = -4$
$a_6 = a_5 - a_4 = -4 - 5 = -9$
$a_7 = a_6 - a_5 = -9 - (-4) = -5$

The first 6 terms are -5, 4, 9, 5, -4, -9 and its sum is 0. The pattern "resets" on the 7th term, i.e., equals the 1st term.
Thus, the sum of terms that are multiples of 6 is 0. I.e., the sum of the first 96 terms is also 0.

That leaves terms 97, 98, 99, and 100, which will be -5, 4, 9, and 5.
The sum of those 4 terms = -5 + 4 + 9 + 5 = 13