Thank you everyone for your comments. I hadn't considered that a square wave would give a different perceived effect than a sine wave which would illuminate the LED for a shorter period at the same frequency.

Indeed.

It's why Ton and Toff are important factors in many applications where frequency becomes less important.

Example, PWM for LED control vs PWM for motor control. The latter does have a liking for specific frequency, whereas LED's (for light) not so much.

 

That is why I repeated it.
The challenge may be in creating a square wave at that power level. I suggest experimenting with a lower voltage because the power in an 18 volt square wave will be quite a bit more than the power in an 18 volt sine wave, or half sine wave, because the voltage below the conduction point of the LED does not deliver any power. And it is the power that causes LED heating, and heat is the enemy.
If that 18 volt transformer is center tapped, then using 2diodes in a ful wave rectifier with a filter should give about 12 volts under load. But if the mains frequency is 50 Hz then the flicker frequency is 50 Hz, because each cycle has both a positive and negative peak. (that is why all of the LEDs lit).