Day 19: Counting Squares

How many perfect squares are there in the infinite set: \[\{1161, 111561, 11115561, 1111155561, \ldots, \underbrace{111\cdots 1}_{n+1}\underbrace{555\cdots 5}_{n-1}61,\ldots\} \] This problem is part of the set Advent Calendar 2014.

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