A052129 a(0) = 1; thereafter a(n) = n*a(n-1)^2.

1, 1, 2, 12, 576, 1658880, 16511297126400, 1908360529573854283038720000, 29134719286683212541013468732221146917416153907200000000

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 A052129(n), then doubles as n-increases to 2n, at which point 1 is added; thus A052129(2n) = 2^n+1. Since 3*A001045(n) = 2^n+1, n-adic valuation of A052129(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.

Links

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

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

Chao-Ping Chen, Sharp inequalities and asymptotic series related to Somos' quadratic recurrence constant, Journal of Number Theory, 172 (2017), 145-159.

Chao-Ping Chen and X. F. Han, On Somos' quadratic recurrence constant, Journal of Number Theory, Volume 166, September 2016, Pages 31-40.

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

Jesús Guillera and Jonathan 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, 155 (2015), 36-45.

Dawei Lu, Xiaoguang Wang, and Ruiqing Xu, Some New Exponential-Function Estimates of the Somos' Quadratic Recurrence Constant, Results in Mathematics 74(1) (2019), Article 6.

Gergo Nemes, On the coefficients of an asymptotic expansion related to Somos' quadratic recurrence constant, Applicable Analysis and Discrete Mathematics, 5(1) (2011), 60-66.

Jörg Neunhäuserer, On the universality of Somos' constant, arXiv:2006.02882 [math.DS], 2020.

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

Jonathan Sondow and Petros 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.

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

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 *)
NestList[{#[[1]]+1, #[[1]]*#[[2]]^2}&, {1, 1}, 10][[All, 2]] (* Harvey P. Dale, Jul 30 2018 *)

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

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