N-growing hierarchy

The N-growing hierarchy is a hierarchy/notation based on the fast-growing hierarchy, created by Japanese googologist Aeton (2013) n-growing hierarchy (Japanese Page).
== Definition ==
* \([m]_0(n) = m\times n\)
* \([m]_{\alpha+1}(n) = [m]^{n-1}_\alpha(m)\), and if \(n=1\), \([m]_{\alpha+1}(1)=[m]^0_\alpha(m)=m\)
* \([m]_\alpha(n) = [m]_{\alpha[n]}(m)\), when \(\alpha\) is a limit ordinal and \(\alpha[n]\) is the \(n\)th term of fundamental sequence assigned to ordinal \(\alpha\).
And when \(m=10\), it can be called 10-growing hierarchy. And similarly, 3-growing hierarchy, 16-growing hierarchy, or Googol-growing hierarchy are also possible.
However, If \(m=n\), it is called Diagonal n-growing hierarchy and its notation changes as follows.
* \((N_\alpha(n) = [n]_\alpha(n))\)
* \(N_0(n) = n\times n=n^2\)
* \(N_{\alpha+1}(n) = N^{n-1}_\alpha(n)\)
* \(N_\alpha(n) = N_{\alpha[n]}(n)\)
=== Examples ===
This function is exactly equal to up-arrow notation, and probably array notation, but for that reason, when \(m=2\) and \(\alpha\geq\omega\), it does not grow well.
* \([16]_4(8) = 16\uparrow^4 8\)
* \([10]_{\omega+1}(100) = \{10,100,1,2\}=\) Corporal
* \([3]^{64}_{\omega}(4)\) = Graham's number \(\lesssim[4]_{\omega+1}(65) = \{4,65,1,2\}\)
* \([4]_{\omega^2+1}(64) = \{4,64,1,1,2\}<\) Fish number 1
* \(N_\omega(3) = [3]_3(3) = 3\uparrow^3 3=\) Tritri
* \(N_{\omega^2}(10) = \{10,10,10,10\}=\) General
Because of the reason that \([m]_{\omega^\omega}(n)=\{m,n+2(1)2\}\), this function doesn't match exactly over \(\{m,n(1)2\}\) level of BEAF, in \(\alpha\geq\omega^\omega\) level.
* \(N_{\omega^{98}}(10) = [10]_{\omega^\omega}(98) = \{10,100 (1) 2\}=\) Goobol
* \([10]_{\omega^\omega}([10]_{\omega^\omega}(98)-2)=\) goobolplex \(\approx[10]^2_{\omega^\omega}(98)\)
== Sources ==
== See also ==
ja:N成長階層
Category:Hierarchies
Category:Functions
Category:Transfinite ordinals
Category:Googology in Asia
Category:Notations