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I had a few 150 Ohm SMD's so I took your suggestion Stan and paralleled them to come up with 75 Ohms. I am using a strip consisting of 12 LED"S for comparison. The circuit with the 75 Ohm set-up was noticeably brighter that the same strip using the 150 Ohm set-up (side by side comparison). So, there was some indication that if I used a larger amount of LED's there may be a "work-around" regarding resistance values using the LM317T regulator. I would love to hear about it.
Annoying to be sure, but there is some math.
You can estimate the current going into the LED strip = 1.25 / resistance. 150 Ohms is 1.25/150 -> 8.3 mA. 75 Ohms -> 16.7 mA.
With 12 LEDs, that's 4 sections of 3-LEDs each. The current is divided between the sections. With 75 Ohms, the 16.7 mA is split 4-ways so each section of 3-LEDs is receiving about 4 mA.
The 3-LED sections can safely handle about 20 mA each so you can crank up the current from where you are with 75 Ohms. But this is just what the LED strip itself can safely handle. As to what the LM317 circuit can safely handle without deforming any nearby plastic is another matter. GRJ provided the relevant math above.
So as you continue to lower the resistance, the LEDs get brighter but the LM317 gets hotter and hotter. At some point, it's simply a better idea to swap out the LM317 with one of the eBay 99-cent (or less) DC-to-DC converter modules. You'd still use the choke, bridge, and capacitor...but replace the LM317/resistor with a DC-to-DC module. This would allow adjustability (small screwdriver) and it would NOT get hot.
Bottom line, if you can get enough brightness using the LM317 and resistor without melting plastic, then you're done. Otherwise, there are (at least) 3 options:
1) Attach the LM317 to a piece of scrap metal to act as a heatsink. This will spread/dissipate the heat. Cost would be as low as a #4 machine-screw and nut to fasten to the hole in the LM317. Note: you cannot directly fasten the LM317 to the metal frame of your car.
2) Add a resistor between the capacitor and the LM317 to dissipate some of the heat. Most economical would be a 1/2 Watt resistor. These run a few pennies each. If it comes to this, one of use will come up with a suitable value after you mess around with what you have to come up with an approximate resistor/brightness that works.
3) Replace the LM317/resistor combo with one of those DC-to-DC eBay regulator modules - the ones that are 99 cents free shipping. You would still use the choke+bridge+capacitor. Advantage here is there's a screwdriver adjustment so you can change brightness.