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Göbel's sequence

{{Short description|Sequence of rational numbers}}

{{refimprove|date=January 2017}}

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, ... {{OEIS|id=A003504}}

The first non-integral value is x43.{{cite book|last1=Guy|first1=Richard K.|title=Unsolved Problems in Number Theory|date=1981|publisher=Springer New York|isbn=978-1-4757-1740-2|page=120}}

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, ... {{OEIS|id=A108394}}

Regardless of the value chosen for k, the initial 19 terms are always integers.{{cite arXiv |eprint=2307.09741 |class= math.NT|title=How long can k-Göbel sequences remain integers? |date=19 July 2023|last1= Matsuhira|first1= Rinnosuke|last2= Matsusaka|first2= Toshiki|last3= Tsuchida|first3= Koki}}{{Cite web |last=Stone |first=Alex |date=2023 |title=The Astonishing Behavior of Recursive Sequences |url=https://www.quantamagazine.org/the-astonishing-behavior-of-recursive-sequences-20231116/ |access-date=2023-11-17 |website=Quanta Magazine |language=en}}

See also

References

{{Reflist}}

External links

  • [http://mathworld.wolfram.com/GoebelsSequence.html Göbel's Sequence]

{{Classes of natural numbers}}

{{DEFAULTSORT:Gobel's sequence}}

Category:Integer sequences

Category:Recurrence relations