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A052129 a(0)=1; thereafter a(n) n*a(n-1)^2. 17
1, 1, 2, 12, 576, 1658880, 16511297126400, 1908360529573854283038720000, 29134719286683212541013468732221146917416153907200000000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Somos's quadratic recurrence sequence.

Iff n is prime (n>2), the n-adic valuation of a(2n) is 3*A001045(n) (three times the values at the prime indices of Jacobsthal numbers), which is 2^n+1. For example: the 11-adic valuation at a(22) = 2049 = 3*A001045(11)= 683.  3*683 = 2^11+1 = 2049.  True because: When n is prime, n-adic valuation is 1 at A520129(n), then doubles as n-increases to 2n, at which point 1 is added; thus A520129(2n) = 2^n+1. Since 3*A001045(n) = 2^n+1, n-adic valuation of A520129(2n) = 3*A001045(n) when n is prime. - Bob Selcoe, Mar 06 2014

Unreduced denominators of: f(1) = 1, f(n) = f(n-1) + f(n-1)/(n-1). - Daniel Suteu, Jul 29 2016

REFERENCES

S. R. Finch, Mathematical Constants, Cambridge University Press, Cambridge, 2003, p. 446.

Xu You, Di-Rong Chen, Improved continued fraction sequence convergent to the Somos' quadratic recurrence constant, Mathematical Analysis and Applications, Volume 436, Issue 1, 1 April 2016, Pages 513-520

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..12

Sung-Hyuk Cha, On the k-ary Tree Combinatorics.

J. Guillera and J. Sondow, Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent, arXiv:math/0506319 [math.NT], 2005-2006.

J. Guillera and J. Sondow, Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent, Ramanujan J. 16 (2008) 247-270.

Dawei Lu and Zexi Song, Some new continued fraction estimates of the Somos' quadratic recurrence constant, Journal of Number Theory, Volume 155, October 2015, Pages 36-45.

J. Sondow and P. Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, arXiv:math/0610499 [math.CA], 2006.

J. Sondow and P. Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, J. Math. Anal. Appl. 332 (2007) 292-314.

Eric Weisstein's World of Mathematics, Somos's Quadratic Recurrence Constant

FORMULA

a(n) ~ s^(2^n) / (n + 2 - 1/n + 4/n^2 - 21/n^3 + 138/n^4 - 1091/n^5 + ...) where s = 1.661687949633... (see A112302) and A116603. - Michael Somos, Apr 02 2006

a(n) = n * A030450(n - 1) if n>0. - Michael Somos, Oct 22 2006

a(n) = (a(n-1) + a(n-2)^2) * (a(n-1) / a(n-2))^2. - Michael Somos, Mar 20 2012

a(n) = product_{k=1..n} k^(2^(n-k)). - Jonathan Sondow, Mar 17 2014

A088679(n+1)/a(n) = n+1. -Daniel Suteu, Jul 29 2016

EXAMPLE

a(3) = 3*a(2)^2 = 3*(2*a(1)^2)^2 = 3*(2*(1*a(0)^2)^2)^2 = 3*(2*(1*1^2)^2)^2 = 3*(2*1)^2 = 3*4 = 12.

G.f. = 1 + x + 2*x^2 + 12*x^3 + 576*x^4 + 1658880*x^5 + 16511297126400*x^6 + ...

MATHEMATICA

Join[{1}, RecurrenceTable[{a[1]==1, a[n]==n a[n-1]^2}, a, {n, 10}]]  (* Harvey P. Dale, Apr 26 2011 *)

a[ n_] := If[ n < 1, Boole[n == 0], Product[ (n - k)^2^k, {k, 0, n - 1}]]; (* Michael Somos, May 24 2013 *)

a[n_] := Product[ k^(2^(n - k)), {k, 1, n}] (* Jonathan Sondow, Mar 17 2014 *)

PROG

(PARI) {a(n) = if( n<1, n==0, prod(k=0, n-1, (n - k)^2^k))}; /* Michael Somos, May 24 2013 */

CROSSREFS

Cf. A000142, A001045, A030450, A112302, A116603, A123851, A123852, A123853, A123854, A238462 (2-adic valuation), A088679.

Sequence in context: A145513 A002860 A108078 * A216335 A173104 A141770

Adjacent sequences:  A052126 A052127 A052128 * A052130 A052131 A052132

KEYWORD

nonn,nice

AUTHOR

Reinhard Zumkeller, Feb 12 2002

STATUS

approved

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Last modified December 11 08:51 EST 2016. Contains 279052 sequences.