Göbel's sequence

In mathematics, a Göbel sequence is a sequence of rational numbers defined by the recurrence relation
 * $$x_n = \frac{ x_0^2+x_1^2+\cdots+x_{n-1}^2}{n-1},\!\,$$

with starting value
 * $$x_0 = x_1 = 1.$$

Göbel's sequence starts with
 * 1, 1, 2, 3, 5, 10, 28, 154, 3520, 1551880, ...

The first non-integral value is x43.

History
This sequence was developed by the German mathematician Fritz Göbel in the 1970s. In 1975, the Dutch mathematician Hendrik Lenstra showed that the 43rd term is not an integer.

Generalization
Göbel's sequence can be generalized to kth powers by
 * $$x_n = \frac{x_0^k+x_1^k+\cdots+x_{n-1}^k}{n}.$$

The least indices at which the k-Göbel sequences assume a non-integral value are
 * 43, 89, 97, 214, 19, 239, 37, 79, 83, 239, ...

Regardless of the value chosen for k, the initial 19 terms are always integers.