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A003504 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)
12
1, 1, 2, 3, 5, 10, 28, 154, 3520, 1551880, 267593772160, 7160642690122633501504, 4661345794146064133843098964919305264116096, 1810678717716933442325741630275004084414865420898591223522682022447438928019172629856 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

REFERENCES

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).

LINKS

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

FORMULA

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

EXAMPLE

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

MAPLE

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

MATHEMATICA

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 *)

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

(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

CROSSREFS

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

Sequence in context: A000617 A132183 A259878 * A213169 A003182 A134294

Adjacent sequences:  A003501 A003502 A003503 * A003505 A003506 A003507

KEYWORD

nonn,easy,nice

AUTHOR

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

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, Dec 12 2007

Further corrections from Max Alekseyev, Mar 04 2009

STATUS

approved

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Last modified December 16 07:41 EST 2017. Contains 296076 sequences.