Editing Ackermann function (section)

== Table of values ==
Computing the Ackermann function can be restated in terms of an infinite table. First, place the natural numbers along the top row. To determine a number in the table, take the number immediately to the left. Then use that number to look up the required number in the column given by that number and one row up. If there is no number to its left, simply look at the column headed "1" in the previous row. Here is a small upper-left portion of the table:

{| class="wikitable"
|+ Values of ''A''(''m'',&nbsp;''n'')
|-
! {{diagonal split header|''m''|''n''}}
! 0
! 1
! 2
! 3
! 4
! ''n''
|-
! 0
| 1 || 2 || 3 || 4 || 5 || <math>n + 1</math>
|-
! 1
| 2 || 3 || 4 || 5 || 6 || <math>n + 2 = 2 + (n + 3) - 3</math>
|-
! 2
| 3 || 5 || 7 || 9 || 11 || <math>2n + 3 = 2\cdot(n + 3) - 3</math>
|-
! 3
| 5 || 13 || 29 || 61 || 125 || <math>2^{(n + 3)} - 3</math>
|- style="vertical-align:top"
! rowspan="3" style="vertical-align:middle" | 4
| rowspan="1" style="border-bottom:0" | 13
| rowspan="1" style="border-bottom:0" | 65533
| rowspan="1" style="border-bottom:0" | 2<sup>65536</sup>&nbsp;−&nbsp;3
| rowspan="1" style="border-bottom:0" | <math>{2^{2^{65536}}} - 3</math>
| rowspan="1" style="border-bottom:0" | <math>{2^{2^{2^{65536}}}} - 3</math>
| rowspan="2" style="border-bottom:0" | <math>\begin{matrix}\underbrace{{2^2}^{{\cdot}^{{\cdot}^{{\cdot}^2}}}}_{n+3} - 3\end{matrix}</math>
|- style="vertical-align:bottom"
| rowspan="2" style="border-top:0" | <math>={2^{2^{2}}}-3</math><br /><math>=2\uparrow\uparrow 3 - 3</math>
| rowspan="2" style="border-top:0" | <math>={2^{2^{2^{2}}}}-3</math><br /><math>=2\uparrow\uparrow 4 - 3</math>
| rowspan="2" style="border-top:0" | <math>={2^{2^{2^{2^{2}}}}}-3</math><br /><math>=2\uparrow\uparrow 5 - 3</math>
| rowspan="2" style="border-top:0" | <math>={2^{2^{2^{2^{2^{2}}}}}}-3</math><br /><math>=2\uparrow\uparrow 6 - 3</math>
| rowspan="2" style="border-top:0" | <math>={2^{2^{2^{2^{2^{2^{2}}}}}}}-3</math><br /><math>=2\uparrow\uparrow 7 - 3</math>
|-
| rowspan="1" style="border-top:0" | <math>=2\uparrow\uparrow (n+3) - 3</math>
|- style="vertical-align:bottom"
! style="vertical-align:middle" | 5
| 65533 <br /><math>=2\uparrow\uparrow(2\uparrow\uparrow 2) - 3</math><br /><math>=2\uparrow\uparrow\uparrow 3 - 3</math>
| <math>2\uparrow\uparrow\uparrow 4 - 3</math>
| <math>2\uparrow\uparrow\uparrow 5 - 3</math>
| <math>2\uparrow\uparrow\uparrow 6 - 3</math>
| <math>2\uparrow\uparrow\uparrow 7 - 3</math>
| <math>2\uparrow\uparrow\uparrow (n+3) - 3</math>
|-
! 6
| <math>2\uparrow\uparrow\uparrow\uparrow 3 - 3</math>
| <math>2\uparrow\uparrow\uparrow\uparrow 4 - 3</math>
| <math>2\uparrow\uparrow\uparrow\uparrow 5 - 3</math>
| <math>2\uparrow\uparrow\uparrow\uparrow 6 - 3</math>
| <math>2\uparrow\uparrow\uparrow\uparrow 7 - 3</math>
| <math>2\uparrow\uparrow\uparrow\uparrow (n+3) - 3</math>
|-
! ''m''
| <math>(2\uparrow^{m-2} 3)-3</math>
| <math>(2\uparrow^{m-2} 4)-3</math>
| <math>(2\uparrow^{m-2} 5)-3</math>
| <math>(2\uparrow^{m-2} 6)-3</math>
| <math>(2\uparrow^{m-2} 7)-3</math>
| <math>(2\uparrow^{m-2}(n+3))-3</math>
|}

The numbers here which are only expressed with recursive exponentiation or [[Knuth's up-arrow notation|Knuth arrows]] are very large and would take up too much space to notate in plain decimal digits.

Despite the large values occurring in this early section of the table, some even larger numbers have been defined, such as [[Graham's number]], which cannot be written with any small number of Knuth arrows. This number is constructed with a technique similar to applying the Ackermann function to itself recursively.

This is a repeat of the above table, but with the values replaced by the relevant expression from the function definition to show the pattern clearly:

{| class="wikitable"
|+ Values of ''A''(''m'',&nbsp;''n'')
|-
! {{diagonal split header|''m''|''n''}}
! 0
! 1
! 2
! 3
! 4
! ''n''
|-
! 0
| 0+1 || 1+1 || 2+1 || 3+1 || 4+1 || ''n'' + 1
|-
! 1
| ''A''(0, 1) || ''A''(0, ''A''(1, 0))<br />= ''A''(0, 2) || ''A''(0, ''A''(1, 1))<br />= ''A''(0, 3) || ''A''(0, ''A''(1, 2))<br />= ''A''(0, 4) || ''A''(0, ''A''(1, 3))<br />= ''A''(0, 5) || ''A''(0, ''A''(1, ''n''−1))
|-
! 2
| ''A''(1, 1) || ''A''(1, ''A''(2, 0))<br />= ''A''(1, 3) || ''A''(1, ''A''(2, 1))<br />= ''A''(1, 5) || ''A''(1, ''A''(2, 2))<br />= ''A''(1, 7) || ''A''(1, ''A''(2, 3))<br />= ''A''(1, 9) || ''A''(1, ''A''(2, ''n''−1))
|-
! 3
| ''A''(2, 1) || ''A''(2, ''A''(3, 0))<br />= ''A''(2, 5) || ''A''(2, ''A''(3, 1))<br />= ''A''(2, 13) || ''A''(2, ''A''(3, 2))<br />= ''A''(2, 29) || ''A''(2, ''A''(3, 3))<br />= ''A''(2, 61) || ''A''(2, ''A''(3, ''n''−1))
|-
! 4
| ''A''(3, 1) || ''A''(3, ''A''(4, 0))<br />= ''A''(3, 13) || ''A''(3, ''A''(4, 1))<br />= ''A''(3, 65533) || ''A''(3, ''A''(4, 2)) || ''A''(3, ''A''(4, 3)) || ''A''(3, ''A''(4, ''n''−1))
|-
! 5
| ''A''(4, 1) || ''A''(4, ''A''(5, 0)) || ''A''(4, ''A''(5, 1)) || ''A''(4, ''A''(5, 2)) || ''A''(4, ''A''(5, 3)) || ''A''(4, ''A''(5, ''n''−1))
|-
! 6
| ''A''(5, 1) || ''A''(5, ''A''(6, 0)) || ''A''(5, ''A''(6, 1)) || ''A''(5, ''A''(6, 2)) || ''A''(5, ''A''(6, 3)) || ''A''(5, ''A''(6, ''n''−1))
|}