If the equation $x^2 = (2^{2^y} + 1)^s + 2^t$ holds, can $t$ be odd?

Question

If $x, y, s, t \in \mathbb{N}$ and the equation $$x^2 = (2^{2^y} + 1)^s + 2^t$$ holds, can $t$ be odd if $s$ is odd?

I know that $t$ cannot be odd when $s=1$. How about if $s>1$? Added September 10 2017: I was mistaken.

Answer 1

Not exhaustive search but i found one example when $s>1$.

for $s=1$ i get many examples.

$s=3,y=2,t=7,x=71$.

for $s>3$ it did not give a small examples, i checked for all $t<100$.