多角形表記
( a ↑↑ b ) ↑↑ n = ( a ↑↑ b ) ↑ { ( a ↑↑ b ) ↑↑ ( n − 1 ) } = [ a ↑ { a ↑ ( b − 1 ) } ] ↑ { ( a ↑↑ b ) ↑↑ ( n − 1 ) } = a ↑ [ { a ↑ ( b − 1 ) } + ( a ↑↑ b ) ↑↑ ( n − 1 ) ] {\displaystyle {\begin{aligned}\left(a\uparrow \uparrow b\right)\uparrow \uparrow n&=\left(a\uparrow \uparrow b\right)\uparrow \left\{\left(a\uparrow \uparrow b\right)\uparrow \uparrow \left(n-1\right)\right\}\\&=\left[a\uparrow \left\{a\uparrow \left(b-1\right)\right\}\right]\uparrow \left\{\left(a\uparrow \uparrow b\right)\uparrow \uparrow \left(n-1\right)\right\}\\&=a\uparrow \left[\left\{a\uparrow \left(b-1\right)\right\}+\left(a\uparrow \uparrow b\right)\uparrow \uparrow \left(n-1\right)\right]\\\end{aligned}}}
a, b が十分に大きければ
a ↑ ( b − 1 ) ≪ ( a ↑↑ b ) ↑↑ ( n − 1 ) {\displaystyle a\uparrow \left(b-1\right)\ll \left(a\uparrow \uparrow b\right)\uparrow \uparrow \left(n-1\right)}