

A270121


Denominators in a perturbed Engel series.


4




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], 20152016, Monatsh. Math. DOI: 10.1007/s0060501508442.


FORMULA

The sequence is generated by taking a(n+1)=b(n1)*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(n1)=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[n1]^2 (1+(n1)a[n1]))/a[n2];
Array[a, 5] (* JeanFrançois 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



