Editing Knuth's up-arrow notation (section)

==Tables of values==
<!-- This section is linked from [[Hyper operator]] -->

===Computing 0↑<sup>''n''</sup>&nbsp;''b''===

Computing <math>0\uparrow^n b = H_{n+2}(0,b) = 0[n+2]b</math> results in
:0, when ''n'' = 0&nbsp; <ref group="nb" name="corona1">Keep in mind that Knuth did not define the operator <math>\uparrow^0</math>.</ref>
:1, when ''n'' = 1 and ''b'' = 0 &nbsp; <ref group="nb" name=corona2>For more details, see [[Exponentiation#Powers of zero|Powers of zero]].</ref><ref group="nb" name=corona3>For more details, see [[Zero to the power of zero]].</ref>
:0, when ''n'' = 1 and ''b'' > 0 &nbsp; <ref group="nb" name=corona2/><ref group="nb" name=corona3/>
:1, when ''n'' > 1 and ''b'' is even (including 0)
:0, when ''n'' > 1 and ''b'' is odd

===Computing 2↑<sup>''n''</sup>&nbsp;''b''===

Computing <math>2\uparrow^n b</math> can be restated in terms of an infinite table. We place the numbers <math>2^b</math> in the top row, and fill the left column with values 2. To determine a number in the table, take the number immediately to the left, then look up the required number in the previous row, at the position given by the number just taken.

{| class="wikitable"
|+ Values of <math>2\uparrow^n b = {} </math> [[Hyperoperation#Notations|<math>H_{n+2}(2,b) = {} </math> <math>2[n+2]b = {} </math>]] [[Conway chained arrow notation|2 → b → n]]
|-
! {{diagonal split header|''ⁿ''|''b''}}
! 1
! 2
! 3
! 4
! 5
! 6
! formula
|-
! 1
| 2 || 4 || 8 || 16 || 32 || 64 || <math>2^b</math>
|-
! 2
| 2 || 4 || 16 || 65,536 || 2,003,...,156,736 || 212,003,...,428,736 || <math>2\uparrow\uparrow b</math>
|-
! 3
| 2 || 4 || 65,536 || 24,636,...,948,736 || 1,300,...,948,736 || 320,146,...,948,736 || <math>2\uparrow\uparrow\uparrow b</math> 
|-
! 4
| 2 || 4 || 24,636,...,948,736 || 68,225,...,948,736 || 167,167,...,948,736 || 3,449,...,948,736 || <math>2\uparrow\uparrow\uparrow\uparrow b</math>
|}

The table is the same as [[Ackermann function#Table of values|that of the Ackermann function]], except for a shift in <math>n</math> and <math>b</math>, and an addition of 3 to all values.

===Computing 3&nbsp;↑<sup>''n''</sup>&nbsp;''b''===

We place the numbers <math>3^b</math> in the top row, and fill the left column with values 3. To determine a number in the table, take the number immediately to the left, then look up the required number in the previous row, at the position given by the number just taken.

{| class="wikitable"
|+ Values of <math>3\uparrow^n b = {} </math> [[Hyperoperation#Notations|<math>H_{n+2}(3,b) = {} </math> <math>3[n+2]b = {} </math>]] [[Conway chained arrow notation|3 → b → n]]
|-
! {{diagonal split header|''ⁿ''|''b''}}
! 1
! 2
! 3
! 4
! 5
! formula
|-
! 1
| 3 || 9 || 27 || 81 || 243 || <math>3^b</math>
|-
! 2
| 3 || 27 || 7,625,597,484,987  || 12,580,...,739,387  ||  338,605,...,355,387 || <math>3\uparrow\uparrow b</math>
|-
! 3
| 3 || 7,625,597,484,987 || 1,945,...,195,387 || 93,652,...,195,387 || 4,854,...,195,387 || <math>3\uparrow\uparrow\uparrow b</math>
|-
! 4
| 3 || 1,945,...,195,387 || 834,215,...,195,387 || 25,653,...,195,387 || 17,124,...,195,387 ||<math>3\uparrow\uparrow\uparrow\uparrow b</math>
|}

===Computing 4 &nbsp;↑<sup>''n''</sup>&nbsp;''b''===

We place the numbers <math>4^b</math> in the top row, and fill the left column with values 4. To determine a number in the table, take the number immediately to the left, then look up the required number in the previous row, at the position given by the number just taken.

{| class="wikitable"
|+ Values of <math>4\uparrow^n b = {} </math> [[Hyperoperation#Notations|<math>H_{n+2}(4,b) = {} </math> <math>4[n+2]b = {} </math>]] [[Conway chained arrow notation|4 → b → n]]
|-
! {{diagonal split header|''ⁿ''|''b''}}
! 1
! 2
! 3
! 4
! 5
! formula
|-
! 1
| 1 || 1 || 1 || 1 || 1 || <math>4^b</math>
|-
! 2
| 1 || 4 || 19,728 ||  603,122,606,263,029,537,... << 19,692 >> ...,149,530,140,391,357,847 ||  10<sup>10<sup>19727</sup></sup> digits || <math>4\uparrow\uparrow b</math>
|-
! 3
| 1 || 12 || 3,638,334,640,024 || 600,225,356,799,454,734,... << 3,638,334,639,988 >> ...,581,273,077,839,447,635 || 10<sup>10<sup>3638334640023</sup></sup> digits || <math>4\uparrow\uparrow\uparrow b</math>
|-
! 4
| 2 || 155 || 807,230,472,602,822,537,... << 118 >> ...,481,244,990,261,351,117 || 10<sup>10<sup>153</sup></sup> digits || 10<sup>10<sup>10<sup>153</sup></sup></sup> digits 
| <math>4\uparrow\uparrow\uparrow\uparrow b</math>
|}

===Computing 10&nbsp;↑<sup>''n''</sup>&nbsp;''b''===

We place the numbers <math>10^b</math> in the top row, and fill the left column with values 10. To determine a number in the table, take the number immediately to the left, then look up the required number in the previous row, at the position given by the number just taken.

{| class="wikitable"
|+ Values of <math>10\uparrow^n b = {}</math> [[Hyperoperation#Notations|<math>H_{n+2}(10,b) = {} </math> <math>10[n+2]b = {} </math>]] [[Conway chained arrow notation|10 → b → n]]
|-
! {{diagonal split header|''ⁿ''|''b''}}
! 1
! 2
! 3
! 4
! 5
! formula
|-
! 1
| 10 || 100 || 1,000 || 10,000 || 100,000 || <math>10^b</math>
|-
! 2
| 10 || 10,000,000,000 || <math>10^{10,000,000,000}</math>  || <math>10^{10^{10,000,000,000}}</math>  || <math>10^{10^{10^{10,000,000,000}}}</math> || <math>10\uparrow\uparrow b</math>
|-
! 3
| 10 || <math>
  \begin{matrix}
   \underbrace{10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}}\\
   10\mbox{ copies of }10
  \end{matrix}</math> || <math>
  \begin{matrix}
   \underbrace{10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}}\\
   \underbrace{10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}}\\
   10\mbox{ copies of }10
  \end{matrix}</math> || <math>
  \begin{matrix}
   \underbrace{10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}}\\
   \underbrace{10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}}\\
   \underbrace{10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}}\\
   10\mbox{ copies of }10
  \end{matrix}</math> || <math>
  \begin{matrix}
   \underbrace{10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}}\\
   \underbrace{10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}}\\
   \underbrace{10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}}\\
   \underbrace{10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}}\\
   10\mbox{ copies of }10
  \end{matrix}</math> || <math>10\uparrow\uparrow\uparrow b</math>
|-
! 4
| 10 || <math>
  \begin{matrix}
   \underbrace{^{^{^{^{^{10}.}.}.}10}10}\\
   10\mbox{ copies of }10
  \end{matrix}</math> || <math>
  \begin{matrix}
   \underbrace{^{^{^{^{^{10}.}.}.}10}10}\\
   \underbrace{^{^{^{^{^{10}.}.}.}10}10}\\
   10\mbox{ copies of }10
  \end{matrix}</math> || <math>
  \begin{matrix}
   \underbrace{^{^{^{^{^{10}.}.}.}10}10}\\
   \underbrace{^{^{^{^{^{10}.}.}.}10}10}\\
   \underbrace{^{^{^{^{^{10}.}.}.}10}10}\\
   10\mbox{ copies of }10
  \end{matrix}</math> || <math>
  \begin{matrix}
   \underbrace{^{^{^{^{^{10}.}.}.}10}10}\\
   \underbrace{^{^{^{^{^{10}.}.}.}10}10}\\
   \underbrace{^{^{^{^{^{10}.}.}.}10}10}\\
   \underbrace{^{^{^{^{^{10}.}.}.}10}10}\\
   10\mbox{ copies of }10
  \end{matrix}</math> 
| <math>10\uparrow\uparrow\uparrow\uparrow b</math>
|}

For 2 ≤ ''b'' ≤ 9 the numerical order of the numbers <math>10\uparrow^n b</math> is the [[lexicographical order]] with ''n'' as the most significant number, so for the numbers of these 8 columns the numerical order is simply line-by-line. The same applies for the numbers in the 97 columns with 3 ≤ ''b'' ≤ 99, and if we start from ''n'' = 1 even for 3 ≤ ''b'' ≤ 9,999,999,999.