I posted the same question in the Drexel Math Forum, and got this result (reply to that post) by Ara M Jamboulian:

Ara'sIdeaexpanded goes like this:

(1) Let a = 8 + sqrt(59) and b = 8 - sqrt(59)

(2) Prove by induction that a^k + b^k is congruent to 6 mod 10 for any integer k >= 2.

Defining and in this way

We see that

and this can be demonstrated via induction.

Let the ordered pair of digits represent the condition that and

Then

, etc.

Since

It follows that

(3) Also prove that floor[a^k]=a^k + b^k - 1

This follows easily since being perpetually less than 1 implies that .

(4) It follows that floor[a^k] is congruent to 5 mod 10.

(Thanks to both the OP and Ara for this interesting problem and help towards its solution.)

I wondered if the numbers 8 and 59 had any special significance. There seems to be none. Some numbers cause the last digit to oscillate through a series of digits. Perhaps there are some surprises in the more general situation, but intuitively I think not.