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A003504
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a(0)=a(1)=1; thereafter a(n+1) = (1/n)*Sum_{k=0..n} a(k)^2 (a(n) is not always integral!).
(Formerly M0728)
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12
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1, 1, 2, 3, 5, 10, 28, 154, 3520, 1551880, 267593772160, 7160642690122633501504, 4661345794146064133843098964919305264116096, 1810678717716933442325741630275004084414865420898591223522682022447438928019172629856
(list;
graph;
refs;
listen;
history;
text;
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OFFSET
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0,3
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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T. D. Noe, Table of n, a(n) for n = 0..16
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. W. Lenstra, Jr., R. K. Guy, and N. J. A. Sloane, Correspondence, 1975-1978
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) [Broken link?]
D. Rusin, Law of small numbers [Broken link]
D. Rusin, Law of small numbers [Cached copy]
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
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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
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EXAMPLE
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a(3) = (1 * 2 + 2^2) / 2 = 3 given a(2) = 2.
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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)
L # Robert FERREOL, Nov 07 2015
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CROSSREFS
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Cf. A005166, A005167, A108394, A115632, A116603 (asymptotic formula).
Sequence in context: A000617 A132183 A259878 * A213169 A330333 A003182
Adjacent sequences: A003501 A003502 A003503 * A003505 A003506 A003507
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane, R. K. Guy
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EXTENSIONS
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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
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STATUS
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approved
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