S function

S function is the fast-growing function defined by Chris Bird. Its growth rate is about \(\theta(\theta_1(\Omega))\) in the fast-growing hierarchy.

Using Bird's Nested Subscript Array Notation, S(n) = \(\{3,n [1 [2 \_{R_n} 2] 2] 2\}\), where:

\(R_i = '1 [1 [2 \_{R_{i-1}} 2] 2] 2'\) (for R>1)

\(R_1 = '1,2'\).

It is trivial to see that S(1) = 3 and S(2) >> H(n), where H(n) denoted Bird's H function.