A270121 Denominators in a perturbed Engel series.

7, 112, 403200, 1755760043520000, 53695136666462381094317154204367872000000

Offset: 1

Author

Andrew Hone, Mar 11 2016

Keywords

• nonn

Comments

The sum of the series 6/a(1)+1/a(2)+1/a(3)+... is a transcendental number, and has a continued fraction expansion whose coefficients are given explicitly in terms of the sequence a(n) and the ratios a(n+1)/a(n).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..8
• Andrew N. W. Hone, Curious continued fractions, nonlinear recurrences and transcendental numbers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.8.4.
• Andrew N. W. Hone, Continued fractions for some transcendental numbers, arXiv:1509.05019 [math.NT], 2015-2016, Monatsh. Math. DOI: 10.1007/s00605-015-0844-2.

Crossrefs

Cf. A112373, A114552, A114550, A270137.

Programs

Mathematica

a[1] = 7; a[2] = 112;
a[n_] := a[n] = (a[n-1]^2 (1+(n-1)a[n-1]))/a[n-2];
Array[a, 5] (* Jean-François Alcover, Dec 16 2018 *)

Formula

The sequence is generated by taking a(n+1)=b(n-1)*a(n)*(1+n*a(n)), b(n)=a(n+1)/a(n) for n>=1 with initial values a(1)=7,b(0)=2. Alternatively, if a(1)=7,a(2)=112 are given then a(n+1)*a(n-1)=a(n)^2*(1+n*a(n)) for n>=2.

Sum_{n>=1} 1/a(n) = -5/7 + A270137. - Amiram Eldar, Nov 20 2020