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A286016 Signed continued fraction expansion with all signs negative of tanh(1).

%I A286016 #40 Jul 02 2022 09:30:12
%S A286016 1,5,2,2,2,2,9,2,2,2,2,2,2,2,2,13,2,2,2,2,2,2,2,2,2,2,2,2,17,2,2,2,2,
%T A286016 2,2,2,2,2,2,2,2,2,2,2,2,21,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,
%U A286016 25,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2
%N A286016 Signed continued fraction expansion with all signs negative of tanh(1).
%C A286016 For any given sequence of signs (e_1, e_2, ..., e_n, ...) one may define the signed continued fraction expansion of a real number x by using floor or ceiling in the step i according to e_i = +1 or e_i = -1. In the present case for the sequence (-1, -1, -1, -1, ...) consisting of only negative signs the ceiling is taken at each step, and the formulas with x_0 = x are a_n = ceiling(x_n) and x_{n+1} = 1/(a_n - x_n). See chapter 1 and 2 of the book by Perron.
%H A286016 Adolf Hurwitz, <a href="http://www.ngzh.ch/archiv/1896_41/41_2/41_26.pdf">Über die Kettenbrüche, deren Teilnenner arithmetische Reihen bilden</a>, Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich, Jahrg. 41 (1896), Jubelband 2, 34-64. Reprinted in Hurwitz's Mathematische Werke.
%H A286016 Oskar Perron, <a href="https://archive.org/details/dielehrevondenk00perrgoog">Die Lehre von den Kettenbrüchen</a>, Teubner, Leipzig, 1930.
%F A286016 Using an obvious condensed notation we get for the sequence 1, 5, 2^(4), 9, 2^(8), 13, 2^(12), 17, 2^(16), 21, 2^(20), ... where 2^(m) means m copies of 2.
%e A286016 a(2) = 5, a(3) = a(4) = a(5) = a(6) = 2, a(7) = 9, etc. These numbers are obtained from the partial quotients xj as follows:
%e A286016 x2 =  (1 +  e^2)/( 2 + 0e^2) ~4.17 so that a(2)=ceiling(x2)=5;
%e A286016 x3 =  (2 + 0e^2)/( 9 - e^2)  ~1.21 so that a(3)=ceiling(x3)=2;
%e A286016 x4 =  (9 -  e^2)/(16 - 2e^2) ~1.31 so that a(4)=ceiling(x4)=2;
%e A286016 x5 = (16 - 2e^2)/(23 - 3e^2) ~1.46 so that a(5)=ceiling(x5)=2;
%e A286016 x6 = (23 - 3e^2)/(30 - 4e^2) ~1.87 so that a(6)=ceiling(x6)=2;
%e A286016 x7 = (30 - 4e^2)/(37 - 5e^2) ~8.11 so that a(7)=ceiling(x7)=9.
%e A286016 The pairs of integers appearing in the xj's are obtained as the principal or as every other of the non-principal approximating fractions of e^2 in the sense of the A. Hurwitz reference.
%p A286016 x:=(exp(1)-exp(-1))/(exp(1)+exp(-1)):b:=ceil(x): x1:=1/(b-x):L:=[b]:
%p A286016 for k from 0 to 40 do:
%p A286016 b1:=ceil(x1): x1:=1/(b1-x1): L:=[op(L),b1]: od: print(L);
%Y A286016 Cf. A004273 (continued fraction of tanh(1)), A280135, A280136.
%K A286016 nonn,cofr
%O A286016 1,2
%A A286016 _Kutlwano Loeto_, Apr 30 2017
%E A286016 More terms from _Jinyuan Wang_, Jul 02 2022