Definition:Göbel's Sequence

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Definition

Göbel's sequence is the sequence defined recursively as:

$x_n = \begin{cases} 1 & : n = 0 \\ \displaystyle \left({1 + \sum_{k \mathop = 0}^{n - 1} {x_k}^2}\right) / n & : n > 0 \end{cases}$


Sequence of Numbers in Göbel's Sequence

Göbel's sequence begins:

$1, 2, 3, 5, 10, 28, 154, 3520, 1 \, 551 \, 880, 267 \, 593 \, 772 \, 160, \ldots$


General Göbel Sequence

Let $m \in \Z_{> 0}$ be a positive integer

The $m$-Göbel sequence is the sequence defined recursively as:

$x_n = \begin{cases} 1 & : n = 0 \\ \displaystyle \left({1 + \sum_{k \mathop = 0}^{n - 1} {x_k}^m}\right) / n & : n > 0 \end{cases}$


Also known as

Göbel's sequence can also be seen rendered as Goebel's sequence.


Historical Note

It is not clear who Göbel's sequence was named for. Research is ongoing.

Some sources link this sequence with the name of Michael Somos, but the latter already has a number of sequences named for him.


Sources