In this post I will compare the following notations:
- FGH
- HH
- SGH
- BEAF
- Bird's Arrays
- Hyp Cos's strong Arrays
- R function
- EXE
- HAN
- My own bracket notation
- AAN
- AM12AN
- My AN
Anything else you think I should add? Let me know.
Below n^n[]
| Standard Form
|
SGH
|
HH
|
FGH
|
BEAF
|
BAN
|
SAN
|
R
|
ExE
|
HAN
|
Bracket
|
AAN
|
AMAN
|
My AN
|
| 0
|
\(g_0(n)\)
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
| a (a number)
|
\(g_a(n)\)
|
a
|
a
|
a
|
a
|
a
|
a
|
a
|
a
|
a
|
a(a)
|
a
|
a
|
| n
|
\(g_\omega(n)\)
|
\(H_0(n)\)
|
n
|
{n}
|
{n}
|
s(n)
|
n
|
n
|
n
|
n
|
a(n)
|
n
|
n
|
| n+a
|
\(g_{\omega+a}(n)\)
|
\(H_a(n)\)
|
\({f_0}^n(n)\)
|
{n}+a
|
{n}+a
|
s(n)+a
|
n+a
|
n+a
|
n+a
|
n+a
|
a(n)+a
|
n+a
|
n+a
|
| 2n
|
\(g_{\omega2}(n)\)
|
\(H_\omega(n)\)
|
\(f_1(n)\)
|
2*{n}
|
2*{n}
|
2*s(n)
|
2n
|
2n
|
2n
|
2n
|
a(n,2)
|
2n
|
n[0]2
|
| 3n
|
\(g_{\omega3}(n)\)
|
\(H_{\omega2}(n)\)
|
3n
|
3{n}
|
3{n}
|
3*s(n)
|
3n
|
3n
|
3n
|
3n
|
a(n,3)
|
3n
|
n[0]3
|
| n^2
|
\(g_{\omega^2}(n)\)
|
n^2
|
n^2
|
{n,2}
|
{n,2}
|
s(n,2)
|
n^2
|
n^2
|
n^2
|
n^2
|
a(n,n)
|
n(n,2)
|
n[1]2 = n[0]n
|
| 2n^2
|
\(g_{\omega^22}(n)\)
|
2n^2
|
2n^2
|
2{n,2}
|
2{n,2}
|
2s(n,2)
|
2n^2
|
2n^2
|
2n^2
|
2n^2
|
a(n,2,1)
|
2*n(n,2)
|
(n[1]2)[0]2
|
| n^3
|
\(g_{\omega^3}(n)\)
|
n^3
|
n^3
|
{n,3}
|
{n,3}
|
s(n,3)
|
n^3
|
n^3
|
n^3
|
n^3
|
a(n,3,1)
|
n(n,3)
|
n[1]3
|
| n^4
|
\(g_{\omega^4}(n)\)
|
n^4
|
n^4
|
{n,4}
|
{n,4}
|
s(n,4)
|
n^4
|
n^4
|
n^4
|
n^4
|
a(n,4,1)
|
n(n,4)
|
n[1]4
|
Up to n^^n[]
| Standard
|
SGH
|
HH
|
FGH
|
BEAF
|
BAN
|
SAN
|
R
|
ExE
|
HAN
|
Bracket
|
AAN
|
AMAN
|
My AN
|
| n^n
|
\(g_{\omega^\omega}(n)\)
|
\(H_{\omega^2}(n)\)
|
\(f_2(n)\)
|
{n,n}
|
{n,n}
|
s(n,n)
|
nR0
|
En
|
n!
|
n|()
|
a(n,n,1)
|
n(n,n)
|
n[1]n
|
| n^(n+1)
|
\(g_{\omega^{\omega+1}}(n)\)
|
\(H_{\omega^2+1}(n)\)
|
\(f_2(f_0(n))\)
|
{n,n+1}
|
{n,n+1}
|
s(n,n+1)
|
(n+1)R0
|
E(n+1)
|
(n+1)!
|
(n+1)|()
|
a(n,n,1)
|
n(n,n+1)
|
n[1](n+1)
|
| n^(2n)
|
\(g_{\omega^{\omega2}}(n)\)
|
\(H_{\omega^2+\omega}(n)\)
|
\(f_2(f_1(n))\)
|
{n,2n}
|
{n,2n}
|
s(n,2n)
|
(2n)R0
|
E(2n)
|
(2n)!
|
(2n)|()
|
a(n,a(n,2),1)
|
n(n,2n)
|
n[1](2n)
|
| n^n^2
|
\(g_{\omega^{\omega^2}}(n)\)
|
\(H_{\omega^2+\omega}(n)\)
|
\(f_2(f_1(n))\)
|
{n,{n,2}}
|
{n,{n,2}}
|
s(n,s(n,2))
|
(n^2)R0
|
E(n^2)
|
(n^2)!
|
(n^2)|()
|
a(n,2,2)
|
n(n,n(n,2)) = n(n,2|2)
|
n[1]n[1]2
|
| n^^3
|
\(g_{\omega^{\omega^\omega}}(n)\)
|
\(H_{\omega^22}(n)\)
|
\(f_2(f_2(n))\)
|
{n,3,2}
|
{n,3,2}
|
s(n,3,2)
|
nR0R0
|
En#2
|
(n!)!
|
n|()|()
|
a(n,3,2)
|
n(n,3|2)
|
n[2]3
|
| n^^4
|
\(g_{\omega^{\omega^{\omega^\omega}}}(n)\)
|
\(H_{\omega^23}(n)\)
|
\(f_2(f_2(f_2(n)))\)
|
{n,4,2}
|
{n,4,2}
|
s(n,4,2)
|
nR0R0R0
|
En#3
|
((n!)!)!
|
n|()|()|()
|
a(n,4,2)
|
n(n,4|2)
|
n[2]4
|
Up to \(f_\omega(n)\)[]
| Up-arrow Notation
|
SGH
|
HH
|
FGH
|
BEAF
|
BAN
|
SAN
|
R
|
ExE
|
HAN
|
| n^^n
|
\(g_{\varepsilon_0}(n)\)
|
\(H_{\omega^3}(n)\)
|
\(f_3(n)\)
|
{n,n,2}
|
{n,n,2}
|
s(n,n,2)
|
nR1
|
En#n
|
n!1
|
| n^^(n+1)
|
\(g_{\omega^{\varepsilon_0+1}}(n)\)
|
\(H_{\omega^3+1}(n)\)
|
\(f_3(f_0(n))\)
|
{n,n+1,2}
|
{n,n+1,2}
|
s(n,n+1,2)
|
(n+1)R1
|
#n#(n+1)
|
(n+1)!1
|
| n^^(2n)
|
\(g_{\varepsilon_1}(n)\)
|
\(H_{\omega^3+\omega}(n)\)
|
\(f_3(f_1(n))\)
|
{n,2n,2}
|
{n,2n,2}
|
s(n,2n,2)
|
(2n)R1
|
En#(2n)
|
(2n)!1
|
| n^^n^2
|
\(g_{\varepsilon_\omega}(n)\)
|
\(H_{\omega^3+\omega*2}(n)\)
|
\(f_3(f_1(n))\)
|
{n,{n,2},2}
|
{n,{n,2},2}
|
s(n,s(n,2),2)
|
(n^2)R1
|
En#(n^2)
|
(n^2)!1
|
| n^^n^n
|
\(g_{\varepsilon_{\omega^\omega}}(n)\)
|
\(h_{\omega^3+\omega^2}(n)\)
|
\(f_3(f_2(n))\)
|
{n,{n,n},2}
|
{n,{n,n},2}
|
s(n,s(n,n),2)
|
nR0R1
|
En#1#2
|
(n!)!1
|
| n^^^3
|
(n)\)
|
\(H_{{\omega^3}2}(n)\)
|
\(f_3(f_3(n))\)
|
{n,3,3}
|
{n,3,3}
|
s(n,3,3)
|
nR1R1
|
En#n#2
|
(n!1)!1
|
| n^^^n
|
\(g_{\zeta_0}(n)\)
|
\(H_{\omega^4}(n)\)
|
\(f_4(n)\)
|
{n,n,3}
|
{n,n,3}
|
s(n,n,3)
|
nR2
|
En#n#n
|
n!2
|
| n^^^n^^n
|
\(g_{\zeta_\varepsilon_0}(n)\)
|
\(H_{\omega^4+\omega^3}(n)\)
|
\(f_4(f_3(n))\)
|
{n,{n,n,2},3}
|
{n,{n,n,2},3}
|
s(n,s(n,n,2),3)
|
nR1R2
|
En#n#1#2
|
(n!1)!2
|
| n^^^^3
|
\(g_{\zeta_\zeta_0}(n)\)
|
\(H_{{\omega^4}2}(n)\)
|
\(f_4(f_4(n))\)
|
{n,3,4}
|
{n,3,4}
|
s(n,3,4)
|
nR2R2
|
En#n#n#2
|
(n!2)!2
|
| n^^^^n
|
\(g_\varphi(3,0)(n)\)
|
\(H_{\omega^5}(n)\)
|
\(f_5(n)\)
|
{n,n,4}
|
{n,n,4}
|
s(n,n,4)
|
nR3
|
En#n#n#n
|
n!3
|
| n^^^^^n
|
\(g_\varphi(4,0)(n)\)
|
\(H_{\omega^6}(n)\)
|
\(f_6(n)\)
|
{n,n,5}
|
{n,n,6}
|
s(n,n,5)
|
nR4
|
En#n#n#n#n
|
n!4
|
Up to \(f_{\omega2}(n)\)[]
