Source code in the jq programming language

# input: [m,n]
def ack:
  .[0] as $m | .[1] as $n
  | if $m == 0 then $n + 1
    elif $n == 0 then [$m-1, 1] | ack
    else [$m-1, ([$m, $n-1 ] | ack)] | ack
    end ;

range(0;5) as $i
| range(0; if $i > 3 then 1 else 6 end) as $j
| "A(\($i),\($j)) = \( [$i,$j] | ack )"

# jq -n -r -f ackermann.jq
A(0,0) = 1
A(0,1) = 2
A(0,2) = 3
A(0,3) = 4
A(0,4) = 5
A(0,5) = 6
A(1,0) = 2
A(1,1) = 3
A(1,2) = 4
A(1,3) = 5
A(1,4) = 6
A(1,5) = 7
A(2,0) = 3
A(2,1) = 5
A(2,2) = 7
A(2,3) = 9
A(2,4) = 11
A(2,5) = 13
A(3,0) = 5
A(3,1) = 13
A(3,2) = 29
A(3,3) = 61
A(3,4) = 125
A(3,5) = 253
A(4,0) = 13


# input: [m,n, cache]
# output [value, updatedCache]
def ack:

  # input: [value,cache]; output: [value, updatedCache]
  def cache(key): .[1] += { (key): .[0] };
  
  def pow2: reduce range(0; .) as $i (1; .*2);
 
  .[0] as $m | .[1] as $n | .[2] as $cache
  | if   $m == 0 then [$n + 1, $cache]
    elif $m == 1 then [$n + 2, $cache]
    elif $m == 2 then [2 * $n + 3, $cache]
    elif $m == 3 then [8 * ($n|pow2) - 3, $cache]
    else
    (.[0:2]|tostring) as $key
    | $cache[$key] as $value
    | if $value then [$value, $cache]
      elif $n == 0 then
        ([$m-1, 1, $cache] | ack)
        | cache($key)
      else
        ([$m, $n-1, $cache ] | ack) 
        | [$m-1, .[0], .[1]] | ack
        | cache($key)
      end
    end;

def A(m;n): [m,n,{}] | ack | .[0];

A(4;1)

65533


A(4;2), A(3;1000000) | tostring | length

19729
301031