Definition:Göbel's Sequence/General
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Definition
Let $m \in \Z_{> 0}$ be a positive integer
The $m$-Göbel sequence is the sequence defined recursively as:
- $x_n = \begin {cases} 1 & : n = 0 \\ \ds \paren {1 + \sum_{k \mathop = 0}^{n - 1} {x_k}^m} / n & : n > 0 \end {cases}$
Examples
$3$-Göbel Sequence
The $3$-Göbel sequence begins:
- $1, 2, 5, 45, 22 \, 815, 2 \, 375 \, 152 \, 056 \, 927, \ldots$
Also known as
Göbel's sequence can also be seen rendered as Goebel's sequence.
Historical Note
Some sources link Göbel's sequence with the name of Michael Somos, but it appears that the latter has a different sequence named for him.
Sources
- Weisstein, Eric W. "Göbel's Sequence." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GoebelsSequence.html