“Göbel's Sequence”的概念、定义、习题解题方法及翻译-数学辞典

单词

Göbel's Sequence

释义

Göbel's Sequence

Consider the Recurrence Relation

(1)

with

. The first few iterates of

are 1, 2, 3, 5, 10, 28, 154, ... (Sloane's A003504). The terms growextremely rapidly, but are given by the asymptotic formula

(2)

where

(3)

(Zagier). It is more convenient to work with the transformed sequence

(4)

which gives the new recurrence

(5)

with initial condition

. Now,

will be nonintegral Iff

. The smallest

for which

(mod

) therefore gives the smallest nonintegral

. In addition, since

is also the smallest nonintegral

For example, we have the sequences :

(6)

(7)

(8)

Testing values of

shows that the first nonintegral

. Note that a direct verification of thisfact is impossible since

(9)

(calculated using the asymptotic formula) is much too large to be computed and stored explicitly.

A sequence even more striking for remaining integral over many terms is the 3-Göbel sequence

(10)

The first few terms of this sequence are 1, 2, 5, 45, 22815, ... (Sloane's A005166).

The Göbel sequences can be generalized to powers by

(11)

See also Somos Sequence

References

Guy, R. K. ``The Strong Law of Small Numbers.'' Amer. Math. Monthly 95, 697-712, 1988.

Guy, R. K. ``A Recursion of Göbel.'' §E15 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 214-215, 1994.

Sloane, N. J. A. SequencesA003504/M0728and A005166/M1551in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

Zaiger, D. ``Solution: Day 5, Problem 3.'' http://www-groups.dcs.st-and.ac.uk/~john/Zagier/Solution5.3.html.

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