How much current will LM317 pass in this circuit?
The key to the circuit is the current going through R1 that gives you the Vref.

Trying to solve this circuit by going the route of using the resistance the LM317 sees will not work, because that would build in a wrong assumption of how the current would flow.

Assuming the current through the ADJ pin is negligible (and the voltages and load are in the regulating range of the LM317), a step by step approach works:

As defined by the operating point of the LM317, \$ V_{R1} = V_{REF} \$:
$$ I_{R1} = \frac{V_{REF}}{R_1} $$

Given \$ I_{R1} = I_{R2} \$:
$$ V_{R2} = I_{R1} R_2 $$

Given \$ V_{R3} = V_{R1} + V_{R2} \$:
$$ I_{R3} = \frac{V_{REF}+V_{R2}}{R_3} $$

$$ I_{OUT} = I_{R1} + I_{R3} $$

It turns out quite straight forward and you can substitute from top to bottom to get a simple formula.