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It turns out that I've slightly misunderstood Pound-Star, because every expression starts with n=1 (and not with the variable expression n=a), you must add a *a to all expressions. (at the end)

Expressions for Pound-Star (#*)

Expression in #* Expression in other notations
n=a, # a
n=a, #*b ab
n=a, #*b*c abc
n=a, #*# a2
n=a, #*#*b a2b2
n=a, #*#*#*b a4b4
n=a, #*#*#*# a8
n=a, #*#...#*# (b+1 #s) a2b
n=a, #*(b) baa
n=a, #*c*(b) baac
n=a, #*#...#*#*(c) (b+1 #s) a2bc2ba
n=a, #*(b)*c bacac
n=a, #*(b)*# ba2a2
n=a, #*(c)*#*#...#*# (b+1 #s) ca2ba2b
n=a, #*(b)*(c) bcaacaa
n=a, #*(b)*(c)*(d) bcdaadaacdaadaa
n=a, #*(2)*(2)...(2)*(2) (b 2s) \(f_{2}^b(a)\) (using FGH, exact)
n=a, #*(b,0) ba+1(a+1)
n=a, #*(b,1) (a+b)a+1(a+1)
n=a, #*(b,c) (ac+b)a+1(a+1)
n=a, #*(b,c,0) ((a+1)c+b)a+2(a+2)
n=a, #*(b,c,d) ((a+1)(c+ad)+b)a+2(a+2)
n=a, #*(b,c,d,e) ((a+2)(c+(a+1)(d+ae))+b)a+3(a+3)

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#*)

Expression in x#* Approximations in other notations

n=a, #*((b))

aa2
n=a, #*((b))*# aa42
n=a, #*((b))*(c) cc2aa3
n=a, #*((b))*((c)) aaa22+2
n=a, #*((b))*((c))*((d)) aaaa22+2
n=a, #*((b))*((b))...((b))*((b)) (c proto-sets) \(a\uparrow\uparrow(c+1)\)

Limit growth rate: \(f_3(x)\).

Approximations for Pound Multistar (#**)

Expressions in #* Approximations in other notations
n=a, #**# aa2
n=a, #**#**# aaa22
n=a, #***# \(a\uparrow\uparrow(a+1)\)
n=a, #***#**# \(a\uparrow\uparrow a^{a^2}\)
n=a, #***#***# \(a\uparrow\uparrow a\uparrow\uparrow(a+1)\)
n=a, #****# \(a\uparrow\uparrow\uparrow(a+1)\)
n=a, #*...*# (b stars) \(a\uparrow^{b-1}(a+1)\)

n=a, #*(b) or

n=a, #**b

baa (exact)

n=a, #*(b)*(c) or

n=a, #**b**c

bcaacaa (exact)

n=a, #*(2)*(2)...(2)*(2) or

n=a, #**2**2...**2**2 (b 2s)

\(f_{2}^b(a)\) (using FGH, exact)

n=a, #**(2) or

n=a, #***2

\(f_{3}(a)\) (using FGH, exact)

n=a, #**...**(2) or

n=a, #**...***2

(b and b+1 stars, respectively)

\(f_{b+1}(a)\) (using FGH, exact)

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: \(f_\omega(x)\).

Expressions 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.

Expressions in H#* FGH
#^2 \(f_{\omega}(a)\) (exact)
#^*2 \(f_{\omega+1}(a)\) (exact)
#^^2 \(f_{\omega2}(a)\) (exact)
#^^^2 \(f_{\omega3}(a)\) (exact)
#^^...^^2 (b carets) \(f_{\omega b}(a)\) (exact)
  1. ^^...^^**...**2

(b carets and c stars)

\(f_{\omega b+c}(a)\) (exact)

Conclusion:

  • The equality with FGH still holds.
  • Limit growth rate: \(f_{\omega^2}(x)\).

Expressions for Exploding Pound Star (#*{})

Expressions in #*{} FGH
#{1}2 \(f_{\omega^2}(a)\) (exact)
#{1}^2 \(f_{\omega^2+\omega}(a)\) (exact)
#{1}^^...^^2 (b carets) \(f_{\omega^2+\omega b}(a)\) (exact)
#{1}{1}2 \(f_{\omega^2 2}(a)\) (exact)
#{1}{1}...{1}{1}2 (b {1}s) \(f_{\omega^2 b}(a)\) (exact)
#{2}2 \(f_{\omega^3}(a)\) (exact)
#{2}{2}...{2}{2}2 (b {1}s) \(f_{\omega^3 b}(a)\) (exact)
#{3}2 \(f_{\omega^4}(a)\) (exact)
#{b}2 \(f_{\omega^{b+1}}(a)\) (exact)

Conclusion:

  • The equality with FGH still holds.
  • Limit growth rate: \(f_{\omega^\omega}(x)\).

Approximations for Hyper Exploding Pound Star (H#*{})

Expressions in H#*{} Approximations in FGH
#{0:1}2 \(f_{\omega^{\omega}}(a+1)\)
#{b:1}2 \(f_{\omega^{\omega+b}}(a)\)
#{0:2}2 \(f_{\omega^{\omega2}}(a)\)
#{0:b}2 \(f_{\omega^{\omega b}}(a)\)
#{0:0:1}2 \(f_{\omega^{\omega^2}}(a)\)
#{0:0:b}2 \(f_{\omega^{\omega^2 b}}(a)\)
#{0:0...0:0:b}2 (c colons) \(f_{\omega^{\omega^c b}}(a)\)

Conclusion:

  • The equality with FGH is lost.
  • Limit growth rate: \(f_{\omega^{\omega^\omega}}(x)\).

Approximations for Extended Hyper Exploding Pound Star (xH#*{})

Expressions in xH#*{} Approximations in FGH
#@2 \(f_{\omega^{\omega^\omega}}(a)\)
#@@2 \(f_{\omega^{\omega^\omega}2}(a)\)
#@....@2 (b @ signs) \(f_{\omega^{\omega^\omega}b}(a)\)

Limit growth rate: \(f_{\omega^{\omega^\omega+1}}(x)\).

Approximations for Nested Hyper Exploding Pound Star (xH#*<>)

Expressions in xH#*<> Approximations in FGH
#@....@2 (b @ signs) \(f_{\omega^{\omega^\omega}b}(a)\)
#<@>2 \(f_{\omega^{\omega^\omega+1}}(a)\)
#<<@>>2 \(f_{\omega^{\omega^\omega+2}}(a)\)
#[1]2 \(f_{\omega^{\omega^\omega+\omega}}(a)\)
#<[1]>2 \(f_{\omega^{\omega^\omega+\omega+1}}(a)\)
#[2]2 \(f_{\omega^{\omega^\omega+\omega2}}(a)\)
#[b]2 \(f_{\omega^{\omega^\omega+\omega b}}(a)\)

Limit growth rate: \(f_{\omega^{\omega^\omega+\omega^2}}(x)\).

Approximations for Multi Nested Hyper Exploding Pound Star (H#*<<>>)

Expressions in H#*<<>> Approximations in FGH
#[]2 \(f_{\omega^{\omega^\omega+\omega^2}}(a)\)
#<[]>2 \(f_{\omega^{\omega^\omega+\omega^2+1}}(a)\)
#[*]2

\(f_{\omega^{\omega^\omega+\omega^2+

\omega}}(a)\)

#[**]2

\(f_{\omega^{\omega^\omega+\omega^2+

\omega2}}(a)\)

#[^]2

\(f_{\omega^{\omega^\omega+\omega^22}}(a)\)

#[{1}]2 \(f_{\omega^{\omega^\omega+\omega^3}}(a)\)
#[{2}]2 \(f_{\omega^{\omega^\omega+\omega^4}}(a)\)
#[{0:1}]2 \(f_{\omega^{\omega^\omega2}}(a)\)
#[{1:1}]2 \(f_{\omega^{\omega^{\omega+1}}}(a)\)
#[{0:0:1}]2 \(f_{\omega^{\omega^{\omega^2}}}(a)\)
#[@]2 \(f_{\omega^{\omega^{\omega^\omega}}}(a)\)
#12 \(f_{\omega^{\omega^{\omega^\omega+\omega}}}(a)\)
#[[]]2 \(f_{\omega^{\omega^{\omega^\omega+\omega^2}}}(a)\)
#[[{0:1}]]2 \(f_{\omega^{\omega^{\omega^\omega2}}}(a)\)
#@2 \(f_{\omega^{\omega^{\omega^{\omega^\omega}}}}(a)\)

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