Fraction of a fraction

Let the nested fraction =y

Then $y=\frac{x-1}{2+y}$

$y^{2}+2y=x-1$

$y^{2}+2y+1=x$

$(y+1)^{2}=x$

Since $(y+1)^{2}$ is an integer, x must be a square number.

The question now is, how many natural numbers up to 2015 are square numbers?

$\sqrt {2015} \approx 44.89$

Rounding down, we get the answer,

$\boxed{44}$