N-growing hierarchy

N-growing hierarchy is one of growing hierarchy-notation based on fast-growing hierarchy, created by Japanese googologist Aeton(2013).

Definition

 * \([m]_0(n) = m\times n\)
 * \([m]_{\alpha+1}(n) = [m]^{n-1}_\alpha(m)\), when n=1, \([m]_{\alpha+1}(1)=[m]^0_\alpha(m)=m\)
 * \([m]_\alpha(n) = [m]_{\alpha[n]}(m)\), when \(\alpha\) is a limit ordinal and \(\alpha[n]\) is the \(n\)th term of fundamental sequence assigned to ordinal \(\alpha\).

And when \(m=10\), This can be called 10-growing hierarchy. And similarly, 3-growing hierarchy 16-growing hierarchy Googol-growing hierarchy are also possible.

However, If \(m=n\), This is called Diagonal n-growing hierarchy and its notation changes as follows.


 * \((N_\alpha(n) = [n]_\alpha(n))\)
 * \(N_0(n) = n\times n\)
 * \(N_{\alpha+1}(n) = N^{n-1}_\alpha(m)\)
 * \(N_\alpha(n) = N_{\alpha[n]}(n)\)

Examples
This function is not approximately equal, but exactly equal to up-arrow notation, and probably array notation, but for that reason, when \(m=2) and \(\alpha\geq\omega\), it don't grows well.


 * \([16]_4(8) = 16\uparrow^4 8\)
 * \([10]_{\omega+1}(100) = {10,100,1,2}\) = Corporal
 * \([3]_{\omega+1}(65) = {3,65,1,2} < G < [4]_{\omega+1}(65) = {4,65,1,2}\)
 * \([4]_{\omega^2+1}(63) = {4,63,1,1,2}\) < Fish Number 1


 * \(N_\omega(3) = [3]_3(3) = 3\uparrow^3 3\) = Tritri
 * \(N_{\omega^2}(10) = \{10,10,10,10\}\) = General


 * \([2]_{\omega}(\infty) = 2\uparrow^\infty 2 = 4\)