0,3

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

One can compute each a(n) for n <= 44 exactly; a(43) has 89242645785 decimal digits. The computation was done in Mathematica. a(43) = 822865(89242645773 digits)286976. - Stan Wagon and Eric W. Weisstein, Oct 11 2025

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).

T. D. Noe, Table of n, a(n) for n = 0..16

Max Alekseyev, Simple recurrence that fails to be integer for the first time at the 44th term, MathOverflow, 2017.

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

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

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

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

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

Cf. A005166, A005167, A097398, A108394, A115632, A116603 (asymptotic formula), A292996.

Cf. A389769 (number of decimal digits in numerator).

Sequence in context: A000617 A132183 A259878 * A213169 A330333 A003182

Adjacent sequences: A003501 A003502 A003503 * A003505 A003506 A003507

nonn,easy,nice

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

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

approved