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, ... Template:OEIS

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, ... Template:OEIS

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

See also

• Somos sequence

External links

• Göbel's Sequence