Odd Squares 7 Less than Nearest Power of 2

Theorem
There exist exactly $3$ odd squares which are $7$ less than the nearest power of $2$:


 * $5^2 = 25 = 2^5 - 7$
 * $11^2 = 121 = 2^7 - 7$
 * $181^2 = 37 \, 761 = 2^{15} - 7$

Note that this sequence includes $1$ and $3$, being all the squares which are $7$ less than a power of $2$.

However, for $1$ and $3$, those powers ($8$ and $16$ respectively) are not the nearest power of $2$ ($1$ and $4$ respectively).