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Göbel's Sequence


Consider the recurrence relation

 x_n=(1+x_0^2+x_1^2+...+x_(n-1)^2)/n,
(1)

with x_0=1. The first few iterates of x_n are 1, 2, 3, 5, 10, 28, 154, ... (OEIS A003504). The terms grow extremely rapidly, but are given by the asymptotic formula

 x_n approx (n+2-n^(-1)+4n^(-2)-21n^(-3)+138n^(-4)-1091n^(-5)+...)C^(2^n)
(2)

(OEIS A116603; correcting Finch 2003, p. 446), where

 C=1.04783144757641122955990946274313755459...
(3)

(OEIS A115632; Finch 2003, p. 446; Zagier). It is more convenient to work with the transformed sequence

 s_n=2+x_1^2+x_2^2+...+x_(n-1)^2=nx_n,
(4)

which gives the new recurrence

 s_(n+1)=s_n+(s_n^2)/(n^2)
(5)

with initial condition s_1=2. Now, s_(n+1) will be nonintegral iff ns_n. The smallest p for which s_p≢0 (mod p) therefore gives the smallest nonintegral s_(p+1). In addition, since ps_p, x_p=s_p/p is also the smallest nonintegral x_p.

For example, we have the sequences {s_n (mod k)}_(n=1)^k:

 2,6=2,5/4=0,0,0  (mod 5)
(6)
 2,6,15=1,5/4=0,0,0,0  (mod 7)
(7)
 2,6,15=4,(52)/9=7,(161)/(16)=8,(264)/5=0,0,...,0  (mod 11).
(8)

Testing values of k shows that the first nonintegral x_n is x_(43). Note that a direct verification of this fact is impossible since

 x_(43) approx 5.4093×10^(178485291567)
(9)

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

A sequence even more striking for assuming integer values only for finitely many terms is the 3-Göbel sequence

 x_n=(1+x_0^3+x_1^3+...+x_(n-1)^3)/n.
(10)

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

The Göbel sequences can be generalized to k powers by

 x_n=(1+x_0^k+x_1^k+...+x_(n-1)^k)/n.
(11)

See also

Somos Sequence

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References

Finch, S. R. Mathematical Constants. Cambridge, England: Cambridge University Press, p. 446, 2003.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. Sequences A003504/M0728, A005166/M1551, A115632, and A116603 in "The On-Line Encyclopedia of Integer Sequences."Zaiger, D. "Solution: Day 5, Problem 3." http://www-groups.dcs.st-and.ac.uk/~john/Zagier/Solution5.3.html.

Referenced on Wolfram|Alpha

Göbel's Sequence

Cite this as:

Weisstein, Eric W. "Göbel's Sequence." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GoebelsSequence.html

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