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%I #46 Nov 06 2021 03:57:37
%S 1,3,1,3,0,3,5,2,8,5,4,9,9,3,3,1,3,0,3,6,3,6,1,6,1,2,4,6,9,3,0,8,4,7,
%T 8,3,2,9,1,2,0,1,3,9,4,1,2,4,0,4,5,2,6,5,5,5,4,3,1,5,2,9,6,7,5,6,7,0,
%U 8,4,2,7,0,4,6,1,8,7,4,3,8,2,6,7,4,6,7,9,2,4,1,4,8,0,8,5,6,3,0,2,9,4,6,7,9
%N Decimal expansion of coth(1).
%C coth(x) = (e^x + e^(-x))/(e^x - e^(-x)).
%C Because the continued fraction for coth(1) is all positive odd numbers in sequence, the second Mathematica program below also generates the sequence. - _Harvey P. Dale_, Oct 15 2011
%C By the Lindemann-Weierstrass theorem, this constant is transcendental. - _Charles R Greathouse IV_, May 14 2019
%D Samuel M. Selby, editor, CRC Basic Mathematical Tables, CRC Press, 1970, p. 218.
%H Ivan Panchenko, <a href="/A073747/b073747.txt">Table of n, a(n) for n = 1..1000</a>
%H Hideyuki Ohtsuka, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.122.7.700">Problem 11853</a>, The American Mathematical Monthly, Vol. 122, No. 7 (2015), p. 700; <a href="https://www.jstor.org/stable/10.4169/amer.math.monthly.124.5.465">A Hyperbolic Sine Series</a>, Solutions to Problem 11853 by Tewodros Amdeberhan and Rituraj Nandan, ibid., Vol. 124, No. 5 (2017), p. 469.
%H Allen Stenger, <a href="https://www.jstor.org/stable/10.4169/amer.math.monthly.124.2.116">Experimental math for Math Monthly problems</a>, The American Mathematical Monthly, Vol. 124, No. 2 (2017), pp. 116-131; <a href="https://www.allenstenger.com/uploads/1/4/1/8/14182140/expmathmathmonthlyfeb2017.pdf">alternative link</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HyperbolicCotangent.html">Hyperbolic Cotangent</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HyperbolicFunctions.html">Hyperbolic Functions</a>.
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F Equals 1 + Sum_{n>=1} (2^(2*n)*B(2*n))/(2*n)! = 1 + Sum_{n>=1} (-1)^(n+1)*2*(A046988(n+1) / A002432(n+1)). - _Terry D. Grant_, May 30 2017
%F Equals 1 + BesselI(3/2, 1)/BesselI(1/2, 1). - _Terry D. Grant_, Jun 18 2018
%F Equals 1 + Sum_{k>=1} csch(2^k) (Ohtsuka, 2015; Stenger, 2017). - _Amiram Eldar_, Oct 04 2021
%e 1.31303528549933130363616124693...
%t RealDigits[Coth[1],10,120][[1]] (* or *) RealDigits[ FromContinuedFraction[ Range[1,1001,2]],10,120][[1]] (* _Harvey P. Dale_, Oct 15 2011 *) (* see Comments, above, for the second program *)
%o (PARI) 1/tanh(1)
%Y Cf. A005408 (continued fraction: odd numbers), A073821 (continued fraction exp. is even numbers), A073744 (tanh(1)=1/A073747), A073742 (sinh(1)), A073743 (cosh(1)), A073745 (csch(1)), A073746 (sech(1)), A349004.
%K cons,nonn
%O 1,2
%A _Rick L. Shepherd_, Aug 07 2002
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