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the broadway marquee flashing lights setupSilly-state is for soul-less geeks.
What classic Broadway did, what even the 1967 Cougar did, is much easier. Get a low-speed motor, "gearmotor", slow enough to make one turn in your full cycle. Put a bunch of disks on the shaft. Get microswitches with roller-levers, one per disk. Jam the switches close so the disk holds them "open", lights off. File notches in the disk so the levers drop and the light comes on. Done.
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The first section, I, IZ, IZZ, IZZO, can be done with four switches.
The later section turns-on 46 lights individually. The dumb solution requires 46 disks and switches. A "simpler" way with a 16-disk motor and a 16P-4T relay is probably worse. Such signs were very rare in real life, perhaps only seen on the Vegas Strip.
Most of the art of sign-flashing is in making the most eye-impress with the least switching. Doing it in PhotoShop is very different, and might be called cheating.
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PIC controller and SS relaysAnother cheat, though very popular. You lose the fun of jammed wheels and rust-bound bearings, you gain some cheap complexity but
46 output relays is a mega job of wiring.
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all it will do is sequencing - not quite your example.The wheels/switch is just a sequencer, and can do what the IZZO animation does, though with far more complexity than simple chase-lights. Put it in position 1. Wire the lights you want on. Put it in position 2, wire the lights for that position. Ah.... everything leaks to everything else. So use DC lights and use isolation diodes so the sneakage only goes the desired way. Ah.... a 4017 can barely lift one LED, so you will need boosters.
It all gets too messy too fast.
I hate to say it, but next year you will be able to get video iPods with cracked cases cheap on eBay. Load it with something more exciting than an old-fashioned sign-flash, and mount it behind your faceplate.