Definition:Göbel's Sequence/General

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