A数列計算表

A数列の計算表です。間違いなどありましたら教えて下さい。
$a\in \mathbb{N} $について$ (a^n) = (\underbrace{a,...,a}_{n個})$です
$ \uparrow $はクヌースの矢印表記です。
$A$ 数列内の数字$a$について$ a\underbrace{,...,}_{n個}a$ を$ a^n$ で略記し、
数字の列$s$と数字$b$について$(s,b\underbrace{,..,}_{n個}b)$を$(s,b)^n$で略記しています。
$n$は十分大きな数とします。
\begin{split} ()[0] &= 0\\ (0)[0] &= ()[0+1]\\ &= ()[1] \\ &=1 \\ (0,0)[0] &= (0)[1] \\ &=()[1+1] \\ &= 2\\ (0^{n})[0] &= ()[n] \\ &= n\\ (0^n)[n] &= ()[n+n]\\ &= 2n\\ (1,1)[n] &= (1,0)^{n}[n]\\ &= (1,0^{n})[n] \\ &= (0^{2n})[2n]\\ &= 2^2n\\ &= 4n \\ (1,1,1)[n] &= (1,1,0^n)[n]\\ &= (1,1)[2n] \\ &= (1,0^{2n})[2n]\\ &= (1)[4n]\\ &= (0^{4n})[4n]\\ &= 2^3n\\ &= 8n \\ (1^n)[n] &= ()[2^nn]\\ &=2^nn　> 2\uparrow n\\ (2)[n] &= (1^n)[n]　> 2\uparrow n\\ (2,1)[n] &= (2,0^n)[n]\\ &= (2)[2n]\\ &= (1^{2n})[2n]\\ &= 2^{2n}\cdot2n > 2\uparrow 2n\\ (2,1,1)[n] &= (2,1,0^n)[n]\\ &= (2,1)[2n]\\ &= (2,0^{2n})[2n]\\ &= (2)[4n]\\ &= (1^{4n})[4n]\\ &= 2^{2^2n}\cdot 2^2n > 2\uparrow 2\uparrow n\\ (2,1^n)[n] &> \underbrace{2\uparrow \cdots \uparrow 2 }_{n個} \uparrow n\\ &> 2\uparrow \uparrow n \uparrow n \\ &\approx 2\uparrow \uparrow n\\ (2,2)[n] &=(2,1^n)[n]　> 2\uparrow \uparrow n\\ (2,2,1)[n] &= (2,2,0^n)[n]\\ &= (2,2)[2n] > 2 \uparrow \uparrow 2n\\ (2,2,1,1)[n] &= (2,2,1,0^n)[n]\\ &= (2,2,1)[2n] \\ &> 2\uparrow \uparrow (2\uparrow 2)n\\ (2,2,1^n)[n] &> 2 \uparrow \uparrow(\underbrace{2\uparrow\cdots\uparrow2}_{n個})n\\ &>2\uparrow\uparrow(2\uparrow\uparrow n)n \\ &> 2\uparrow\uparrow(2\uparrow\uparrow n)\\ (2,2,2)[n] &= (2,2,1^n)[n]　> 2\uparrow\uparrow(2\uparrow\uparrow n+1)\\ (2,2,2,2)[n] &= (2,2,2,1^n)[n] > 2\uparrow\uparrow(2\uparrow\uparrow(2\uparrow\uparrow n+1) )\\ (2^n)[n] &> \underbrace{2\uparrow\uparrow\cdots\uparrow\uparrow2}_{n個}\uparrow\uparrow n\\ &\approx 2\uparrow\uparrow\uparrow n =2\uparrow^3 n\\ (3)[n] &= (2^n)[n] > 2\uparrow^3 n\\ (3,1)[n] &= (3)[2n] > 2\uparrow^3 2n\\ (3,1^n)[n] &= (3)[2\uparrow n] > 2\uparrow^3 (2\uparrow n)\\ (3,2)[n] &= (3,1^n)[n] > 2\uparrow^3 (2\uparrow n)\\ (3,2,2)[n] &= (3,2)[2\uparrow n] > 2\uparrow^3 (2\uparrow 2\uparrow n)\\ (3,2^n)[n] &> 2\uparrow^3(2\uparrow^2 n)+2 > 2\uparrow^3(2\uparrow^2 n)\\ (3,3)[n] &= (3,2^n)[n] > 2\uparrow^3(2\uparrow^2 n)\\ (3,3,3)[n] &=(3,3,2^n)[n]\\ &> (3,3)[2\uparrow^3 n] \\ &> 2\uparrow^3(2\uparrow^2 2\uparrow^3 n) \\ &> 2\uparrow^3 2\uparrow^3n\\ (4)[n] &= (3^n)[n]\\ &> \underbrace{2\uparrow^3\cdots\uparrow^3 2}_{n個}\uparrow^3 n\\ &\approx 2\uparrow^4 n\\ (5)[n] &> 2\uparrow^5 n\\ (m)[n] &> 2\uparrow^m n\\ (n)[n] &> 2\uparrow^n n \approx f_{\omega}(n)\\ \end{split}
また$ A(n,n) $をアッカーマン関数として
$ (n)[n] > A(n,n) $