User blog:Wythagoras/Pound Star Analysis

Expressions in Pound-Star (#*)
Conclusion:
 * Multiple numbers give rise to exponential growth.
 * Multiple pound signs give rise to double exponential growth.
 * Multiple sets give rise to tetrational growth.
 * One set with k entries gives rise to exponential growth.
 * Limit growth rate: \(f_3(x)\).

Approximations for Extended Pound Star (x#*)
Limit growth rate: \(f_3(x)\).

Approximations for Pound Multistar (#**)
For other sets and the proto sets the growth rate is almost the same as for the simple set (2).

Conclusion:
 * Multiple stars work like the hyper operators.
 * Pound-Star statistifies the nice equation (n=a, #**...**(2)) = (n=a, #**...***2) = \(f_{b+1}(a)\). (b and b+1 stars, respectively)
 * Limit growth rate: \(f_\omega(x)\).

Approximations for Hyper Pound Star (H#*)
From now on, we'll only look to expressions with a 2 at the end because that relates exactly to FGH and the growth rate is similiar for other expressions at the end. Also, the expression n=a is left out. Conclusion:
 * The equality with FGH still holds.
 * Limit growth rate: \(f_{\omega^2}(x)\).

Approximations for Exploding Pound Star (#*{})
Conclusion:
 * The equality with FGH still holds.
 * Limit growth rate: \(f_{\omega^\omega}(x)\).