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*'''n*<sup>[k,x#<sub>n[n,<sub>o</sub>m-1]<sub>o-1</sub>n,<sub>o</sub>m-1</sub>|n]</sup> = n*<sup>[k,x#<sub>1,<sub>o</sub>m-1</sub>|2]</sup>''' |
*'''n*<sup>[k,x#<sub>n[n,<sub>o</sub>m-1]<sub>o-1</sub>n,<sub>o</sub>m-1</sub>|n]</sup> = n*<sup>[k,x#<sub>1,<sub>o</sub>m-1</sub>|2]</sup>''' |
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*'''n*<sup>[k,x#<sub>n[n[n,<sub>o</sub>m-1]<sub>o-1</sub>n,<sub>o</sub>m-1]<sub>o-1</sub>n,<sub>o</sub>m-1</sub>|n]</sup> = n*<sup>[k,x#<sub>1,<sub>o</sub>m-1</sub>|3]</sup>''' |
*'''n*<sup>[k,x#<sub>n[n[n,<sub>o</sub>m-1]<sub>o-1</sub>n,<sub>o</sub>m-1]<sub>o-1</sub>n,<sub>o</sub>m-1</sub>|n]</sup> = n*<sup>[k,x#<sub>1,<sub>o</sub>m-1</sub>|3]</sup>''' |
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| + | |||
| − | --> |
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| + | === Eighth Extension === |
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| + | ...There |
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| + | |||
| + | === Ninth Extension === |
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| + | ...is |
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| + | |||
| + | === Tenth Extension === |
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| + | ...no |
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| + | |||
| + | === Eleventh Extension === |
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| + | ...extension |
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| + | |||
| + | Don't uncomment this.--> |
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==Sources== |
==Sources== |
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Revision as of 18:37, 26 November 2013
Mixed factorial \(n^*\) is a function recursively defined as
\[1^* = 1\]
\[(n + 1)^* = n^* +^n (n + 1)\]
where \(+^n\) is the \(n\)th hyper operator, starting at addition. For example, \(4^* = ((1 + 2) \cdot 3) \uparrow 4\). Informally, the sequence can be visualized as starting with 1, adding 2, multiplying by 3, exponentiating by 4, tetrating by 5, ...
The function was coined by an author under the alias of "SpongeTechX".
Sources
See Also
Main article: Factorial
Multifactorials: Double factorial · MultifactorialFalling and rising: Falling factorial · Rising factorial
Other mathematical variants: Alternating factorial · Hyperfactorial · q-factorial · Roman factorial · Subfactorial · Weak factorial · Primorial · Compositorial · Semiprimorial
Tetrational growth: Exponential factorial · Expostfacto function · Superfactorial by Clifford Pickover
Nested Factorials: Tetorial · Petorial · Ectorial · Zettorial · Yottorial
Array-based extensions: Hyperfactorial array notation · Nested factorial notation
Other googological variants: · Tetrofactorial · Superfactorial by Sloane and Plouffe · Torian · Factorexation · Mixed factorial · Bouncing Factorial
Main article: Mixed Factorial
Basic: Tetrafact · Pentafact · Hexafact · CentifactSeries 2: Centifact · Double Centifact · Triple Centifact · Quadruple Centifact · Quintuple Centifact · Decuple Centifact · Centuple Centifact
Series 3: Hypercentifact · Duo-hypercentifact · Ultracentifact