The Primality of

Consider numbers $t (n)$ of the form $t (n) = 2 n^{2} - 1$ with $n > 1$ .
The first such numbers are $7, 17, 31, 49, 71, 97, 127$ and $161$ .
It turns out that only $49 = 7 \cdot 7$ and $161 = 7 \cdot 23$ are not prime.
For $n \leq 10000$ there are $2202$ numbers $t (n)$ that are prime.

How many numbers $t (n)$ are prime for $n \leq 50 000 000$ ?

Write-up coming later

The complete problem is available here. An approach, code, and answer will be added later.