Problem on Elliptic Curve Point Doubling
The point addition $P+Q$ and doubling $2P = P +P $ in Elliptic Curves $E$ are not just x,y coordinates in the Euclidean Plane that you can add the coordinates. One can find the rules in Wikipedia;

Point addition: With 2 distinct points, P and Q, addition is defined as the negation of the point resulting from the intersection of the curve, E, and the straight line defined by the points P and Q, giving the point, R.
Point doubling: Where the points P and Q, are coincident (at the same coordinates), addition is similar, except that there is no well-defined straight line through P and Q, so the operation is closed using limiting case, the tangent to the curve, E, at P and Q.


Given the Elliptic curve $E:y^2= x^3+a x + b$ and a point $P=(x_p,y_p)$ on the curve, the doubling $R=2P = P + P$ can be calculated by:
\begin{align}
\lambda &amp;= \frac{3 x_p^2 + a }{2 y_p}\\
x_r &amp;= \lambda^2 -2 x_p\\
y_r &amp;= \lambda(x_p-x_r) - y_p
\end{align}
The curve is defined in modular arithmetic $\pmod{17}$, therefore convert $1/2$ into $2^{-1} \equiv 9 \bmod 17$ and find the inverse by extended GCD algorithm.
$\lambda = (3\cdot 5^2 +2)\cdot 9) \equiv 13 \bmod 17$
$x_r = \lambda^2 - 2 x_p  = (13^2  -10 ) \equiv 6 \bmod 17 $
$x_y = \lambda(x_p - x_r)-y_p =  13 \cdot ( 1 - 6) - 1 \equiv 3 \bmod 17$
$$(5,1) + (5,1) = (6,3)$$