A003504 a(0)=a(1)=1; thereafter a(n+1) = (1/n)*Sum_{k=0..n} a(k)^2 (a(n) is not always integral!).

1, 1, 2, 3, 5, 10, 28, 154, 3520, 1551880, 267593772160, 7160642690122633501504, 4661345794146064133843098964919305264116096, 1810678717716933442325741630275004084414865420898591223522682022447438928019172629856

Offset: 0

Author

N. J. A. Sloane, R. K. Guy

Keywords

• nonn
• easy
• nice

Comments

The sequence appears with a different offset in some other sources. - Michael Somos, Apr 02 2006

Also known as Göbel's (or Goebel's) Sequence. Asymptotically, a(n) ~ n*C^(2^n) where C=1.0478... (A115632). A more precise asymptotic formula is given in A116603. - M. F. Hasler, Dec 12 2007

Let s(n) = (n-1)*a(n). By considering the p-adic representation of s(n) for primes p=2,3,...,43, one finds that a(44) is the first nonintegral value in this sequence. Furthermore, for n>44, the valuation of s(n) w.r.t. 43 is -2^(n-44), implying that both s(n) and a(n) are nonintegral. - M. F. Hasler and Max Alekseyev, Mar 03 2009

a(44) is approximately 5.4093*10^178485291567. - Hans Havermann, Nov 14 2017.

The fractional part is simply 24/43 (see page 709 of Guy (1988)).

The more precise asymptotic formula is a(n+1) ~ C^(2^n) * (n + 2 - 1/n + 4/n^2 - 21/n^3 + 138/n^4 - 1091/n^5 + ...). - Michael Somos, Mar 17 2012

Examples

			a(3) = (1 * 2 + 2^2) / 2 = 3 given a(2) = 2.
		

References

• R. K. Guy, Unsolved Problems in Number Theory, 3rd edition, Sect. E15.
• Clifford Pickover, A Passion for Mathematics, 2005.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 0..16
• Hibiki Gima, Toshiki Matsusaka, Taichi Miyazaki, and Shunta Yara, On integrality and asymptotic behavior of the (k,l)-Göbel sequences, arXiv:2402.09064 [math.NT], 2024. See p. 1.
• R. K. Guy, Letter to N. J. A. Sloane, Sep 25 1986.
• R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.
• R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
• H. Ibstedt, Some sequences of large integers, Fibonacci Quart. 28 (1990), 200-203.
• Yuh Kobayashi and Shin-ichiro Seki, On the length over which k-Göbel sequences remain integers, arXiv:2502.17448 [math.CO], 2025.
• H. W. Lenstra, Jr., R. K. Guy, and N. J. A. Sloane, Correspondence, 1975-1978
• N. Lygeros and M. Mizony, Study of primality of terms of a_k(n)=(1+(sum from 1 to n-1)(a_k(i)^k))/(n-1)
• Rinnosuke Matsuhira, Toshiki Matsusaka, and Koki Tsuchida, How long can k-Göbel sequences remain integers?, arXiv:2307.09741 [math.NT], 2023.
• D. Rusin, Law of small numbers [Broken link]
• D. Rusin, Law of small numbers [Cached copy]
• Alex Stone, The Astonishing Behavior of Recursive Sequences, Quanta Magazine, Nov 16 2023, 13 pages.
• Eric Weisstein's World of Mathematics, Göbel's Sequence
• D. Zagier, Problems posed at the St Andrews Colloquium, 1996
• D. Zagier, Solution: Day 5, problem 3

Crossrefs

Cf. A005166, A005167, A097398, A108394, A115632, A116603 (asymptotic formula).

Programs

Maple

a:=2: L:=1,1,a: n:=15: for k to n-2 do a:=a*(a+k)/(k+1): L:=L,a od:L; # Robert FERREOL, Nov 07 2015

Mathematica

a[n_] := a[n] = Sum[a[k]^2, {k, 0, n-1}]/(n-1); a[0] = a[1] = 1; Table[a[n], {n, 0, 13}] (* Jean-François Alcover, Feb 06 2013 *)
With[{n = 14}, Nest[Append[#, (#.#)/(Length[#] - 1)] &, {1, 1}, n - 2]] (* Jan Mangaldan, Mar 21 2013 *)

PARI

A003504(n,s=2)=if(n-->0,for(k=1,n-1,s+=(s/k)^2);s/n,1) \\ M. F. Hasler, Dec 12 2007

Python

a=2; L=[1,1,a]; n=15
for k in range(1,n-1):
    a=a*(a+k)//(k+1)
    L.append(a)
print(L) # Robert FERREOL, Nov 07 2015

Formula

a(n+1) = ((n-1) * a(n) + a(n)^2) / n if n > 1. - Michael Somos, Apr 02 2006

0 = a(n)*(+a(n)*(a(n+1) - a(n+2)) - a(n+1) - a(n+1)^2) +a(n+1)*(a(n+1)^2 - a(n+2)) if n>1. - Michael Somos, Jul 25 2016

Extensions

a(0)..a(43) are integral, but from a(44) onwards every term is nonintegral - H. W. Lenstra, Jr.

Corrected and extended by M. F. Hasler, Dec 12 2007

Further corrections from Max Alekseyev, Mar 04 2009