A270121 Denominators in a perturbed Engel series.

7, 112, 403200, 1755760043520000, 53695136666462381094317154204367872000000

Offset

1,1

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.

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

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 *)

Crossrefs

Cf. A112373, A114552, A114550, A270137.

Sequence in context: A010795 A293456 A099153 * A079296 A081531 A142537

Adjacent sequences: A270118 A270119 A270120 * A270122 A270123 A270124

Keyword

nonn

Author

Andrew Hone, Mar 11 2016

Status

approved