User blog:Googleaarex/Extended Up Arrow Notation

We defined \(a \uparrow_2 b\), which is equal to \(a \uparrow ... \uparrow a\) (b \(\uparrow\)'s).

Then \(a \uparrow\uparrow_2 b\) = \(a \uparrow_2 a ... a \uparrow_2 a\) (b a's), \(a \uparrow\uparrow\uparrow_2 b\) = \(a \uparrow\uparrow_2 a ... a \uparrow\uparrow_2 a\) (b a's), etc.

Next, \(a \uparrow_2\uparrow_2 b\) = \(a \uparrow ... \uparrow\uparrow_2 a\) (b \(\uparrow\)'s), \(a \uparrow_2\uparrow_2\uparrow_2 b\) = \(a \uparrow ... \uparrow\uparrow_2\uparrow_2 a\) (b \(\uparrow\)'s), etc.

The next arrow type is \(\uparrow_3\). \(a \uparrow_3 b\) = \(a \uparrow_2 ... \uparrow_2 a\) (b \(\uparrow_2\)'s). Then \(\uparrow_4\), \(\uparrow_6\), etc.