|
%I M0728 #96 Feb 17 2024 10:26:45
%S 1,1,2,3,5,10,28,154,3520,1551880,267593772160,7160642690122633501504,
%T 4661345794146064133843098964919305264116096,
%U 1810678717716933442325741630275004084414865420898591223522682022447438928019172629856
%N a(0)=a(1)=1; thereafter a(n+1) = (1/n)*Sum_{k=0..n} a(k)^2 (a(n) is not always integral!).
%C The sequence appears with a different offset in some other sources. - _Michael Somos_, Apr 02 2006
%C 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
%C 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
%C a(44) is approximately 5.4093*10^178485291567. - _Hans Havermann_, Nov 14 2017.
%C The fractional part is simply 24/43 (see page 709 of Guy (1988)).
%C 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
%D R. K. Guy, Unsolved Problems in Number Theory, 3rd edition, Sect. E15.
%D Clifford Pickover, A Passion for Mathematics, 2005.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A003504/b003504.txt">Table of n, a(n) for n = 0..16</a>
%H Hibiki Gima, Toshiki Matsusaka, Taichi Miyazaki, and Shunta Yara, <a href="https://arxiv.org/abs/2402.09064">On integrality and asymptotic behavior of the (k,l)-Göbel sequences</a>, arXiv:2402.09064 [math.NT], 2024. See p. 1.
%H R. K. Guy, <a href="/A005169/a005169_6.pdf">Letter to N. J. A. Sloane</a>, Sep 25 1986.
%H R. K. Guy, <a href="http://www.jstor.org/stable/2322249">The strong law of small numbers</a>. Amer. Math. Monthly 95 (1988), no. 8, 697-712.
%H R. K. Guy, <a href="/A005165/a005165.pdf">The strong law of small numbers</a>. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
%H H. Ibstedt, <a href="http://www.fq.math.ca/Scanned/28-3/ibstedt.pdf">Some sequences of large integers</a>, Fibonacci Quart. 28 (1990), 200-203
%H H. W. Lenstra, Jr., R. K. Guy, and N. J. A. Sloane, <a href="/A003504/a003504.pdf">Correspondence, 1975-1978</a>
%H N. Lygeros and M. Mizony, <a href="https://web.archive.org/web/20050310052154/http://igd.univ-lyon1.fr/home/mizony/premiers.html">Study of primality of terms of a_k(n)=(1+(sum from 1 to n-1)(a_k(i)^k))/(n-1)</a>
%H Rinnosuke Matsuhira, Toshiki Matsusaka, and Koki Tsuchida, <a href="https://arxiv.org/abs/2307.09741">How long can k-Göbel sequences remain integers?</a>, arXiv:2307.09741 [math.NT], 2023.
%H D. Rusin, <a href="http://www.math.niu.edu/~rusin/known-math/96/smallnums">Law of small numbers</a> [Broken link]
%H D. Rusin, <a href="/A003504/a003504.txt">Law of small numbers</a> [Cached copy]
%H Alex Stone, <a href="https://www.quantamagazine.org/the-astonishing-behavior-of-recursive-sequences-20231116/">The Astonishing Behavior of Recursive Sequences</a>, Quanta Magazine, Nov 16 2023, 13 pages.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GoebelsSequence.html">Göbel's Sequence</a>
%H D. Zagier, <a href="http://www-groups.mcs.st-andrews.ac.uk/~john/Zagier/Problems.html">Problems posed at the St Andrews Colloquium, 1996</a>
%H D. Zagier, <a href="http://www-groups.mcs.st-andrews.ac.uk/~john/Zagier/Solution5.3.html">Solution: Day 5, problem 3</a>
%F a(n+1) = ((n-1) * a(n) + a(n)^2) / n if n > 1. - _Michael Somos_, Apr 02 2006
%F 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
%e a(3) = (1 * 2 + 2^2) / 2 = 3 given a(2) = 2.
%p 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
%t 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 *)
%t With[{n = 14}, Nest[Append[#, (#.#)/(Length[#] - 1)] &, {1, 1}, n - 2]] (* _Jan Mangaldan_, Mar 21 2013 *)
%o (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
%o (Python)
%o a=2; L=[1,1,a]; n=15
%o for k in range(1,n-1):
%o ....a=a*(a+k)//(k+1)
%o ....L.append(a)
%o L # _Robert FERREOL_, Nov 07 2015
%Y Cf. A005166, A005167, A097398, A108394, A115632, A116603 (asymptotic formula).
%K nonn,easy,nice
%O 0,3
%A _N. J. A. Sloane_, _R. K. Guy_
%E a(0)..a(43) are integral, but from a(44) onwards every term is nonintegral - H. W. Lenstra, Jr.
%E Corrected and extended by _M. F. Hasler_, Dec 12 2007
%E Further corrections from _Max Alekseyev_, Mar 04 2009
| 