User blog comment:B1mb0w/Growth Rate of the S Function/@comment-25601061-20160708203814/@comment-5529393-20160713045738

I agree with Alemagno. I'm getting:

S(n,S(T(0),0,1),1) = f_{w+1}(n)

S(n,S(T(0),0,m),1) = f_{w+m}(n)

S(n,S(T(0),1,1),1) = f_{w2}(n)

S(n,S(T(0),2,1),1) = f_{w^2}(n)

S(n,S(T(0),3,1),1) = f_{e0}(n)

S(n, T(1), 1) = S(n,S(T(0),T(0),1),1) = f_{phi(w,0)}(n)

S(n, T(1), 2) = f_{phi(w,0)}^2(n)

S(n, S(T(1),0,1), 1) = f_{phi(w,0)+1}(n)

S(n, S(T(1),1,1), 1) = f_{phi(w,0)2}(n)

S(n, S(T(1),n+2,1), 1) = f_{phi(n, phi(w,0)+1)}(n)

S(n, S(T(1),T(0), 1), 1) = f_{phi(w,1)}(n)

S(n, S(T(1), S(T(0),0,1), 1), 1) = S(n, S(T(1), T(0), T(1)), 1) = f_{phi(w,phi(w,0))}(n)

S(n, S(T(1), S(T(0),0,2), 1), 1) = f_{phi(w+1, phi(w,0))}(n)

S(n, S(T(1), S(T(0),1,1), 1), 1) = f_{phi(w2, 0)}(n)

S(n, S(T(1), S(T(0),2,1), 1), 1) = f_{phi(w^2,0)}(n)

S(n, S(T(1), S(T(0),3,1), 1), 1) = f_{phi(epsilon_0,0)}(n)

S(n, T(2), 1) = S(n, S(T(1), T(1), 1), 1) = f_{phi(phi(w,0), 0)}(n)

S(n, T(3), 1) = f_{phi(phi(phi(w,0), 0), 0)}(n)