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A123168 Continued fraction for c = sqrt(2)*(exp(sqrt(2))-1)/(exp(sqrt(2))+1).

%I A123168 #48 Oct 05 2023 08:38:00
%S A123168 0,1,6,5,14,9,22,13,30,17,38,21,46,25,54,29,62,33,70,37,78,41,86,45,
%T A123168 94,49,102,53,110,57,118,61,126,65,134,69,142,73,150,77,158,81,166,85,
%U A123168 174,89,182,93,190,97,198,101,206,105,214,109,222,113
%N A123168 Continued fraction for c = sqrt(2)*(exp(sqrt(2))-1)/(exp(sqrt(2))+1).
%C A123168 This continued fraction shows exp(sqrt(2)) is irrational (see A274540).
%D A123168 J. Borwein and D. Bailey, Mathematics by experiment, plausible reasoning in the 21st Century, A. K. Peters, p. 77.
%H A123168 G. C. Greubel, <a href="/A123168/b123168.txt">Table of n, a(n) for n = 1..1500</a>
%H A123168 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,-1).
%F A123168 a(2*n) = 4*n-3, a(2*n+1) = 8*n-2.
%F A123168 From _Colin Barker_, Apr 15 2012: (Start)
%F A123168 a(n) = 2*a(n-2) - a(n-4) for n>5.
%F A123168 G.f.: x^2*(1+6*x+3*x^2+2*x^3)/((1-x)^2*(1+x)^2). (End)
%F A123168 a(n) = (2*n-3)*(3-(-1)^n)/2 for n>1, with a(1) = 0. - _Wesley Ivan Hurt_, Apr 01 2022
%t A123168 $MinPrecision = 5 $MachinePrecision; ContinuedFraction[Sqrt[2]* (Exp[Sqrt[2]] - 1)/(Exp[Sqrt[2]] + 1), 100]  (* _G. C. Greubel_, Aug 17 2018 *) (* or *)
%t A123168 LinearRecurrence[{0, 2, 0, -1}, {0, 1, 6, 5, 14}, 100] (* _Georg Fischer_, Aug 26 2022 *)
%o A123168 (PARI) default(realprecision, 1000); contfrac(sqrt(2)*(exp(sqrt(2))-1)/ (exp(sqrt(2))+1)) \\ _Michel Marcus_, Oct 11 2016
%Y A123168 Cf. A000217, A000384, A212343.
%Y A123168 Odd bisection of A062828 with 0 prepended.
%Y A123168 Cf. A274540, A123251.
%K A123168 nonn,cofr,easy
%O A123168 1,3
%A A123168 _Benoit Cloitre_, Oct 02 2006