2015 HMMT November General Round Problem 10

Let N be the number of functions f from \{1,2,\ldots,101\} \to \{1,2,\ldots,101\} such that f^{101}(1) = 2. Find the remainder when N is divided by 103.

Answers are checked by value, so any equivalent form is accepted: 1/2, \frac{1}{2} and 0.5 all count as the same answer.

Full credit goes to HMMT for authoring these problems. This problem is from the November 2015 contest; the official solution is available on the HMMT archive. HMMT is not affiliated with or endorsing TopsOJ in any way.