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A270121 Denominators in a perturbed Engel series. 4

%I #20 Nov 20 2020 06:52:23

%S 7,112,403200,1755760043520000,

%T 53695136666462381094317154204367872000000

%N Denominators in a perturbed Engel series.

%C 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).

%H Amiram Eldar, <a href="/A270121/b270121.txt">Table of n, a(n) for n = 1..8</a>

%H Andrew N. W. Hone, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Hone/hone3.html">Curious continued fractions, nonlinear recurrences and transcendental numbers</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.8.4.

%H Andrew N. W. Hone, <a href="http://arxiv.org/abs/1509.05019">Continued fractions for some transcendental numbers</a>, arXiv:1509.05019 [math.NT], 2015-2016, Monatsh. Math. DOI: 10.1007/s00605-015-0844-2.

%F 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.

%F Sum_{n>=1} 1/a(n) = -5/7 + A270137. - _Amiram Eldar_, Nov 20 2020

%t a[1] = 7; a[2] = 112;

%t a[n_] := a[n] = (a[n-1]^2 (1+(n-1)a[n-1]))/a[n-2];

%t Array[a, 5] (* _Jean-François Alcover_, Dec 16 2018 *)

%Y Cf. A112373, A114552, A114550, A270137.

%K nonn

%O 1,1

%A _Andrew Hone_, Mar 11 2016

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Last modified March 28 12:26 EDT 2024. Contains 371254 sequences. (Running on oeis4.)