LM317 Application Notes: LED Constant Current Source Scheme
(Excerpt) The LM317L wants to see 1.25 V between its VOUT pin and the Vadj pin, and it will do whatever it can to keep that voltage differential between them.
But what if a series/parallel combination of LEDs is wanted in the LM317 circuit? The following circuit works fine, assuming that there are not large variations in the forward voltage drop of the LEDs. There could be a problem however if one of the strings opens up for some reason. The LEDs that remain would have 50% more current flowing through them, which could cause them to be driven too hard and fail. So how does one get around this problem? Download LM317 Application Note from On Semi.