A four-day signal

A strategy watches a stock whose daily moves are up (U) or down (D) with probability $1/2$ each, independently from day to day. It fires on any day when the last four moves read U, U, D, U. Two firings may share days, as in UUDUUDU, which fires twice.

(a) Over 252 trading days, what is the expected number of days on which it fires?

(b) Starting from day 1, what is the expected number of days until it first fires?

(c) Right after it fires, what is the expected number of days until it fires again? Why does this differ from (b)?