Английская Википедия:Göbel's sequence
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In mathematics, a Göbel sequence is a sequence of rational numbers defined by the recurrence relation
- <math>x_n = \frac{ x_0^2+x_1^2+\cdots+x_{n-1}^2}{n-1},\!\,</math>
with starting value
- <math>x_0 = x_1 = 1.</math>
Göbel's sequence starts with
- 1, 1, 2, 3, 5, 10, 28, 154, 3520, 1551880, ... Шаблон:OEIS
The first non-integral value is x43.[1]
History
This sequence was developed by the German mathematician Fritz Göbel in the 1970s.[2] In 1975, the Dutch mathematician Hendrik Lenstra showed that the 43rd term is not an integer.[2]
Generalization
Göbel's sequence can be generalized to kth powers by
- <math>x_n = \frac{x_0^k+x_1^k+\cdots+x_{n-1}^k}{n}.</math>
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, ... Шаблон:OEIS
Regardless of the value chosen for k, the initial 19 terms are always integers.[3][2]
See also
References
External links
Шаблон:Classes of natural numbers
