[seqfan] Multiplying waterfalls

[Eric Angelini]

Hello SeqFans,
Start S with a(1) > 9
Now a(n) is simply the result of a multiplication :
[the (n-1)th digit of S] x [the n-th digit of S]

Let's start S respectively with a(1) = [10, 11, 12, 13, 14, 15, 16, ...]

S = 10,0,0,0,0,0,0,0,...

S = 11,1,1,1,1,1,1,1,...
                                                    __
S = 12,2,4,8,32,24,6,4,8,24,24,32,16,8,8,8,12,6,2,6,48,64,64,8,2,12,12,12,24,32,...
                                                   __
S = 13,3,9,27,18,14,7,8,8,4,28,56,64,32,8,16,40,30,36,24,12,6,16,8,6,24,0,0,0,0,18,...
                                                          __
S = 14,4,16,4,6,24,12,8,4,2,16,32,8,2,6,18,6,16,16,12,6,8,48,6,6,6,6,6,2,12,48,32,32,...

S = 15,5,25,10,10,5,0,0,0,0,0,0,0,0,...
                          __
S = 16,6,36,18,18,6,8,8,8,48,48,64,64,32,...
                                        __
(the "overlined" multiplications --like 48-- produce the last computed so far term
 of some of the above sequences)

Question:
What are the integers >9 which end on a fixed point (a fixed point like 0 or 1)
or end in a loop? Respectively, what are the integers escaping fixed points and loops?
Though a 2-digit multiplication cannot produce a term > 81, I've noticed that 13,
for instance, never ends in a loop -- because the first "0,0,0,0" pattern will
turn later in a "0,0,0,0,0" (5-zeroes) pattern, which will turn later in a
6-zeroes pattern, than 7-zeroes, etc. -- and between those 0-patterns you will
always count more and more digits...

Best,
É.