Writing first $5$ terms of recurrence relation

Question

How do I write the first $5$ terms for this recurrence relation?

$$S_0=2\\ S_n=S_{n-1}^2 +S_{n-2}^2+\ldots+S_0^2$$

Since I can't substitute $S_0$ directly?

Answer 1

What you are meant to do is: $$S_0 = 2\\ S_1 = S_0^2 = 2^2 = 4 \\ S_2 = S_1^2+S_0^2 = 4^2+2^2 = 20 \\ S_3 = S_2^2+S_1^2+S_0^2 = 400 + 16 + 4 = 420 $$ And so forth (two more lines) until you have $S_5$..