%I A003504 M0728
%S A003504 1,1,2,3,5,10,28,154,3520,1551880,267593772160,7160642690122633501504,
%T A003504 4661345794146064133843098964919305264116096,
%U A003504 1810678717716933442325741630275004084414865420898591223522682022447438928019172629856
%N A003504 a(0)=a(1)=1; thereafter a(n+1) = sum(a(k)^2,k=0..n)/n (a(n) is not always 
               integral!).
%C A003504 Also known as Gobel'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 A003504 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 A. Alekseyev, Mar 03 2009)
%D A003504 R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), 
               no. 8, 697-712.
%D A003504 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A003504 T. D. Noe, <a href="b003504.txt">Table of n, a(n) for n=0..16</a>
%H A003504 N. Lygeros & M. Mizony, <a href="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 A003504 D. Rusin, <a href="http://www.math.niu.edu/~rusin/known-math/96/smallnums">
               Law of small numbers</a>
%H A003504 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               GoebelsSequence.html">Link to a section of The World of Mathematics.</
               a>
%H A003504 D. Zagier, <a href="http://www-groups.dcs.st-and.ac.uk/~john/Zagier/Problems.html">
               Problems posed at the St Andrews Colloquium, 1996</a>
%H A003504 D. Zagier, <a href="http://www-groups.dcs.st-and.ac.uk./~john/Zagier/
               Solution5.3.html">Solution: Day 5, problem 3</a>
%F A003504 a(n+1) = ((n-1)*a(n)+a(n)^2)/n
%o A003504 (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
%Y A003504 Cf. A005166, A005167, A108394, A115632, A116603 (asymptotic formula).
%Y A003504 Sequence in context: A088938 A000617 A132183 this_sequence A003182 A134294 
               A154956
%Y A003504 Adjacent sequences: A003501 A003502 A003503 this_sequence A003505 A003506 
               A003507
%K A003504 nonn,easy,nice
%O A003504 0,3
%A A003504 N. J. A. Sloane (njas(AT)research.att.com), R. K. Guy
%E A003504 a(0)..a(43) are integral, but from a(44) onwards every term is nonintegral 
               - H. W. Lenstra, Jr.
%E A003504 Corrected and extended by M. F. Hasler (maximilian.hasler(AT)gmail.com), 
               Dec 12 2007
%E A003504 Further corrections from Max Alekseyev (maxale(AT)gmail.com), Mar 04 
               2009