In fact,

ɸⁿ + (-ɸ)⁻ⁿ = F(n + 1) + F(n - 1)

where F(n) is the Fibonacci sequence (F(0) = 0, F(1) = 1, F(n + 1) = F(n) + F(n - 1)).

So the closest integer to ɸⁿ is F(n + 1) + F(n - 1), and the distance between ɸⁿ and the closest integer is ɸ⁻ⁿ.

To prove this, you can first prove the following identities by induction on n:

ɸⁿ = F(n) ɸ + F(n - 1)

(-ɸ)⁻ⁿ = F(n + 1) - F(n) ɸ

There are a lot of algebraic numbers with this property. For example:

The Nth power of [sqrt(3) + sqrt(2)] ** 2 becomes close-to-integral as N approaches infinity.

That’s because [sqrt(3) + sqrt(2)] ** 2N + [sqrt(3) - sqrt(2)] ** 2N is exactly integral. Due to the alternating signs, the irrational terms cancel out. Since [sqrt(3) - sqrt(2)] ** 2N goes to zero as N gets large, the observed behavior follows.

Pi is transcendental so no relationship like this exists and the apparently ergodic behavior isn’t surprising.