If A=(3+sqrt(5))/2 and B=(3-sqrt(5))/2 then

S(n) = (-2)(-1)^n + A^n + B^n. This explicit formula suggests that all terms of the sequence are perfect squares.

More to the point, I believe that the sequence is equivalent to

a(1), a(2), a(3), . . . where a(1)=a(2)=1 and

sqrt(a(n+2)) = sqrt(a(n+1)) + sqrt(a(n)) which clearly has all perfect square terms.

I realize that I have left out details - my approach is too long and cumbersome to write in this format.