$$
\begin{align}
P_{average} &= 1/T\times \int^T_0{P}dt\quad\left(\mbox{where T is the period, }2\pi/\omega\right)\\
&= \frac{V_0^2}{RT}\left[\frac{t}{2} - \frac{1}{4\omega}\sin(2\omega t)\right]^T_0 = \frac{V_0^2}{2R}
\end{align}
$$
The DC power formula for voltage is $P = V^2/R$, so the good representative (RMS) voltage is $V_0/\sqrt{2}\;.$
If you draw out the power graph for the bulb, the frequency of the power is twice the frequency of the voltage, and the graph sits just above the $P=0$ line. The graph of the light output will follow this frequency, but will be raised up on the $y$-axis by some fixed amount, as of course the filament stays hot between cycles! There will also be a shift to the right because the filament will keep
heating beyond the peak power for a short time.