Prove that $\sum_{k=0}^{ \frac{n^2}{2}} 2^{n^2-2k}\binom{n^2}{2k} = \frac{3^{n^2}+1}{2}$

Question

Question is for even n>0. I found this to be true, even wolfram says so, but when I click on show steps it won't.

Question again: $$\sum_{k=0}^{ \frac{n^2}{2}} 2^{n^2-2k}\binom{n^2}{2k} = \frac{3^{n^2}+1}{2}$$

Answer 1

Hint: try $\frac{(2+1)^{n^{2}}+(2-1)^{n^{2}}}{2}$