[seqfan] Multiplication by themselves of the last k digits

[Eric Angelini]

Hello SeqFans,

Start S with any integer;
Extend S with the result of the multiplication by themselves
                                  of the last 2 digits of S;
S always "ends" like this:
- 0,0,0,0,0,0,0,0,0,... or
- 1,1,1,1,1,1,1,1,1,... or
- 2,4,8,32,6,12,(2),...

Now:
Start T with any integer;
Extend T with the result of the multiplication by themselves
                                  of the last 3 digits of T;
T always "ends" like this:
- 0,0,0,0,0,0,0,0,0,... or
- 1,1,1,1,1,1,1,1,1,... or
- 16,48,192,18,(16),...

I didn't check U (multiplication of the last 4 digits), etc.
Is there a sequence somewhere? [this might be old hat but
I couldn't find anything in the OEIS]
Best,
É.

P.-S.
To start a seq with a single digit is easy: just multiply
the said digit by himself - and so on.