About: Ackermann function   Sponge Permalink

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The function grows at similar rates as the much less known Sudan function. As an example, the expansion of A(2,2) is given below: \[ \begin{array}{cclclcl} A(2,2) &=& A(1,A(2,1))\\ &=& A(1,A(1,A(2,0)))&&\\ &=& A(1,A(1,A(1,1)))\\ &=& A(1,A(1,A(0,A(1,0))))\\ &=& A(1,A(1,A(0,A(0,1))))\\ &=& A(1,A(1,A(0,2)))\\&=& A(1, A(1, 3))\\ &=& A(1, A(0, A(1, 2)))\\ &=& A(1, A(0, A(0, A(1, 1))))\\ &=& A(1, A(0, A(0, A(0, A(1, 0)))))\\ &=& A(1, A(0, A(0, A(0, A(0, 1)))))\\ &=& A(1, A(0, A(0, A(0, 2))))\\ &=& A(1, A(0, A(0, 3)))\\ &=& A(1, A(0, 4))\\ &=& A(1, 5)\\ &=& A(0, A(1, 4))\\ &=& A(0, A(0, A(1, 3)))\\ &=& A(0, A(0, A(0, A(1, 2))))\\ &=& A(0, A(0, A(0, A(0, A(1, 1)))))\\ &=& A(0, A(0, A(0, A(0, A(0, A(1, 0))))))\\ &=& A(0, A(0, A(0, A(0, A(0, A(0, 1))))))\\ &=& A(0, A(0, A(0, A(0, A(0, 2)))))\\ &=& A

AttributesValues
rdfs:label
  • Ackermann function
rdfs:comment
  • The function grows at similar rates as the much less known Sudan function. As an example, the expansion of A(2,2) is given below: \[ \begin{array}{cclclcl} A(2,2) &=& A(1,A(2,1))\\ &=& A(1,A(1,A(2,0)))&&\\ &=& A(1,A(1,A(1,1)))\\ &=& A(1,A(1,A(0,A(1,0))))\\ &=& A(1,A(1,A(0,A(0,1))))\\ &=& A(1,A(1,A(0,2)))\\&=& A(1, A(1, 3))\\ &=& A(1, A(0, A(1, 2)))\\ &=& A(1, A(0, A(0, A(1, 1))))\\ &=& A(1, A(0, A(0, A(0, A(1, 0)))))\\ &=& A(1, A(0, A(0, A(0, A(0, 1)))))\\ &=& A(1, A(0, A(0, A(0, 2))))\\ &=& A(1, A(0, A(0, 3)))\\ &=& A(1, A(0, 4))\\ &=& A(1, 5)\\ &=& A(0, A(1, 4))\\ &=& A(0, A(0, A(1, 3)))\\ &=& A(0, A(0, A(0, A(1, 2))))\\ &=& A(0, A(0, A(0, A(0, A(1, 1)))))\\ &=& A(0, A(0, A(0, A(0, A(0, A(1, 0))))))\\ &=& A(0, A(0, A(0, A(0, A(0, A(0, 1))))))\\ &=& A(0, A(0, A(0, A(0, A(0, 2)))))\\ &=& A
dcterms:subject
dbkwik:googology/p...iPageUsesTemplate
fgh
  • \omega
Base
  • Successor function
abstract
  • The function grows at similar rates as the much less known Sudan function. As an example, the expansion of A(2,2) is given below: \[ \begin{array}{cclclcl} A(2,2) &=& A(1,A(2,1))\\ &=& A(1,A(1,A(2,0)))&&\\ &=& A(1,A(1,A(1,1)))\\ &=& A(1,A(1,A(0,A(1,0))))\\ &=& A(1,A(1,A(0,A(0,1))))\\ &=& A(1,A(1,A(0,2)))\\&=& A(1, A(1, 3))\\ &=& A(1, A(0, A(1, 2)))\\ &=& A(1, A(0, A(0, A(1, 1))))\\ &=& A(1, A(0, A(0, A(0, A(1, 0)))))\\ &=& A(1, A(0, A(0, A(0, A(0, 1)))))\\ &=& A(1, A(0, A(0, A(0, 2))))\\ &=& A(1, A(0, A(0, 3)))\\ &=& A(1, A(0, 4))\\ &=& A(1, 5)\\ &=& A(0, A(1, 4))\\ &=& A(0, A(0, A(1, 3)))\\ &=& A(0, A(0, A(0, A(1, 2))))\\ &=& A(0, A(0, A(0, A(0, A(1, 1)))))\\ &=& A(0, A(0, A(0, A(0, A(0, A(1, 0))))))\\ &=& A(0, A(0, A(0, A(0, A(0, A(0, 1))))))\\ &=& A(0, A(0, A(0, A(0, A(0, 2)))))\\ &=& A(0, A(0, A(0, A(0, 3))))\\ &=& A(0, A(0, A(0, 4)))\\ &=& A(0, A(0, 5))\\ &=& A(0, 6)\\ &=& 7 \end{array} \] Using Knuth's up-arrow notation, \(A(x,y) = 2\uparrow^{x-2}(y+3)-3\).
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