User blog comment:ArtismScrub/how fast is this?/@comment-30754445-20171227173219

How did you get two different results for n![4,4]?

Seems straightforward to me:

n![4,4] = (n![3,4])![4,3] = ...

What other option is their?

At any rate, I get:

n![2,2] = n![4]

n![3,3] = n![18]

n![4,4] = n![87]

n![5,5] = n![449]

In general, n![k,k] grows roughly like f3(k^k)... but this is a really misleading way to look at what you've done.

Note that niether n nor the factorial are playing any role when expanding your arrays. What you've actually created is a notation [a,b] that your using to count the factorials. This notation has exponential growth: [3,3] = 18, [4,4] = 87, [5,5] = 449... which is pretty good for a 2-entry array. It just seems slow because your comparing it to the repeated factorials which grow faster, but that's a very misleading comparision to make.

So give this notation a chance and see where it goes. It might surprise you. At the very least, try to define [a,b,c]. I'm pretty sure that n![k,k,k] will be at least f3(f3(k)) and it might turn out to grow even faster.