A familiar sequence

In the above circuit, each resistor has a resistance of 1 \(\Omega\) and the wires have no resistance.

As the number of resistors approaches infinity, the equivalent resistance approaches \(\dfrac{a + \sqrt{b}} { c},\) where \(a\), \(b\), and \(c\) are positive integers, and \(b\) is square free. What is \(a + b + c\)?

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