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A003504
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a(0)=a(1)=1; thereafter a(n+1) = sum(a(k)^2,k=0..n)/n (a(n) is not always integral!).
(Formerly M0728)
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9
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1, 1, 2, 3, 5, 10, 28, 154, 3520, 1551880, 267593772160, 7160642690122633501504, 4661345794146064133843098964919305264116096, 1810678717716933442325741630275004084414865420898591223522682022447438928019172629856
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| 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
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)
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REFERENCES
| R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..16
N. Lygeros & M. Mizony, Study of primality of terms of a_k(n)=(1+(sum from 1 to n-1)(a_k(i)^k))/(n-1)
D. Rusin, Law of small numbers
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
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FORMULA
| a(n+1) = ((n-1)*a(n)+a(n)^2)/n
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PROG
| (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
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CROSSREFS
| Cf. A005166, A005167, A108394, A115632, A116603 (asymptotic formula).
Sequence in context: A088938 A000617 A132183 * A003182 A134294 A154956
Adjacent sequences: A003501 A003502 A003503 * A003505 A003506 A003507
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), R. K. Guy
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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 (maximilian.hasler(AT)gmail.com), Dec 12 2007
Further corrections from Max Alekseyev (maxale(AT)gmail.com), Mar 04 2009
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