User:Wythagoras/Rado's sigma function/BB(5,5)

\(\Sigma(5,5) > 2\uparrow\uparrow(1.7 \cdot 10^{352})\) 0 _ 1 l 1 0 1 2 r 0 0 2 1 l 0 0 3 2 r 1 0 4 2 r 0 1 _ _ l 0 1 1 2 r 1 1 2 3 r 1 1 3 4 r 0 1 4 1 l 4 2 _ 1 r 2 2 1 _ l 2 2 2 3 l 3 2 3 1 l 4 2 4 1 r halt 3 1 3 l 3 3 _ 1 l 2 4 4 4 l 4 4 3 3 l 4 4 2 2 l 4 4 1 1 l 4 4 _ 1 l 2

Explanation
The first part is the machine for \(\Sigma(2,5)\). It outputs: 222...22241 (~1.7*10^352 2's)         ^

Then state 4 brings the head to the beginning of the string of two's, and adds a one. Then for each 3 the number of ones will be duplicated, and for and for each 2 all the ones will be changed into threes, giving a function comparable to \(2^n\). It uses LittlePeng9's duplication machine.