Definition:Göbel's Sequence

Definition

Göbel's 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}^2} / 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 \\ \ds \paren {1 + \sum_{k \mathop = 0}^{n - 1} {x_k}^m} / n & : n > 0 \end {cases}$

Also known as

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

Source of Name

This entry was named for Frits Göbel.

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

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $43$

• Weisstein, Eric W. "Göbel's Sequence." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GoebelsSequence.html