Fish number 1

Fish number 1 starts with the Ackermann function.

After that, we get a \(S_2\) function that maps functions of \(\S_1\) and in general, we have a \(S_n\) function that maps functions of \(\S_{n-1}\): The Fish function \(F_1(x) = S_x(x,x)\) grows about as fast as \(f_{\omega^2}(x)\). The Fish function is similiar to Taro's multivariable Ackermann function. In fact, \(S_z(x,y) = A(x,y,z)\).
 * \(S_1(0,y) = y+1\)
 * \(S_1(x,0) = S_1(x - 1,1)\)
 * \(S_1(x,y) = S_1(x - 1,S_1(x,y - 1))\)
 * \(S_1(0,y) = y+1\)
 * \(S_z(0,y) = S_{z-1}(y,y)\) for \(z ≥ 1\)
 * \(S_z(x,0) = S_z(x - 1,1)\)
 * \(S_z(x,y) = S_z(x - 1,S_z(x,y - 1))\)

The Fish number 1 is defined as \(F_1^63(3)\). Fish number 1 is comparable to \(A(1,0,1,63)\)