# Definition:Göbel's Sequence

## Contents

## 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

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

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