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