I have trouble following math in Wikipedia's solution for the twin paradox. In particular, I don't understand how they derived coordinate velocity $v(t)=\frac{at}{\sqrt{1+\left(\frac{at}{c}\right)^2}}$.
Based on my understanding, it seems like $\frac{a}{\sqrt{1+\left(\frac{at}{c}\right)^2}}$ is coordinate acceleration and $at$ is probably proper velocity (?). I am not sure why the square root part is $1+\left(\frac{at}{c}\right)^2$, not $1-\left(\frac{at}{c}\right)^2$.
To understand where your equation comes from you need to understand the coordinate transformation for an observer in constant proper acceleration. This is derived in many (but not all!) books on relativity. I first encountered it in chapter 6 of Gravitation by Misner, Thorne and Wheeler.
I won't go through the derivation here since you can read it in detail in the book, but the result is given in equation 6.5 on page 166 (in the first edition). In the rest frame of the accelerating observer they experience a constant proper acceleration, i.e. "g force", $a$ and their clock shows time $\tau$. Their position is of course constant at zero since in their rest frame they are at rest at the origin. If both observers start at the origin at time $t = \tau = 0$ then the coordinates of the accelerating observer in the frame of the inertial observer as a function of proper time are:
$$\begin{align} t &= \frac{c}{a} \sinh\left(\frac{a\tau}{c}\right) \\ x &= \frac{c^2}{a} \cosh\left(\frac{a\tau}{c}\right) - \frac{c^2}{a} \end{align}$$
Now we can get the velocity in the inertial observer's frame using:
$$v = \frac{dx}{dt} = \frac{dx}{d\tau} \frac{d\tau}{dt} = c\tanh\left(\frac{a\tau}{c}\right) $$
and finally we use the identity:
$$ \tanh(\operatorname{arcsinh}(x)) = \frac{x}{\sqrt{1+x^2}} $$
to get:
$$ v = \frac{at}{\sqrt{1 + (at/c)^2}} $$
which is the equation given in the Wikipedia article.
Note that in this last equation $v$ and $t$ are the coordinate velocity and time in the frame of the inertial observer while $a$ is the proper acceleration not the coordinate acceleration. The coordinate acceleration is given by $a/\gamma^3$.
You may be interested to read Phil Gibbs's article on the relativistic rocket since it gives this and many other equations, though without derivation.
