Part of your description is a bit garbled, but here is what I think you are asking: You need to control three lights. Each will go thru a separate pattern with everything repeating in 15 second periods. You have a 1 Hz square wave that is to be used as the reference clock.
You mention one implementation method using muxes and logic gates, but give no justification for why you think that is the best answer or any constraints why they need to be used, so I'll just ignore that.
The obvious answer is a microcontroller. Even the smallest cheapest of them all, the PIC 10F200 can do this. It has 4 I/O pins and can be configured to 1 input and 3 outputs, which is exactly what you need. It can also generate its own timing, so you could ignore the 1 Hz input if you wanted to and use interal timing.
In either case, this is a very simple program. Nothing changes faster than on 500 ms boundaries. Since the whole pattern repeats every 15 seconds, you have only 30 possible different states. A very simple and also robust firmware implementation is to have a table of all 30 states. Each 500 ms clock edge, the firmware advances to the next table entry and copies the 3 bit value to the output pins. This is very simple, but allows for easy modification of the display pattern as requirements evolve because only the table entries need to change. The logic stays the same. A 10F200 has 255 usable instructions, which is plenty for the simple clock edge detect, table lookup, output update, and then the table itself.