I would imagine this is a task best left to the analog domain, not the digital domain. Whatever you produce would be a poor substitute for an AC waveform, and limited in terms of available frequencies.
That said, the DC square wave mentioned in the instructable can just be a PWM output set to a 50% duty cycle at the right frequency.
You don't actually need anything else but a square wave to drive the coil as the resonance will turn it into a sin wave. That instructables circuit doesn't need to two 10R resistors across the supply just put the other end of the coil to ground and switch between + and ground with the transistors. Derive the square wave from either the Arduino or the NE555 as shown. You will get about 200V across the coil with a 5V supply when you hit resonance. Tune for resonance by adjusting the capacitor and tweaking the frequency.
Just treat it like a motor changing direction very quickly, that is send the current down the coil one way and then send it down the other. Use a H bridge circuit, just like a motor.