two fast-growing sequences

[Jeremy Gardiner]

I have been investigating some mathematical structures that give rise to the
following algorithm and corresponding sequences:

b=0
do
   a := 2^b
   b := a^2
loop

A(n) = 1, 2, 16, 1.157921E+77, ...

B(n) = 0, 1, 4, 256, 1.340781E+154, ...

The terms grow too rapidly for me to calculate further: does anyone
recognise these sequences?

Thanks,
Jeremy Gardiner