A073747 Decimal expansion of coth(1).

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, 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, 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

Offset

1,2

Comments

coth(x) = (e^x + e^(-x))/(e^x - e^(-x)).

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

By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 14 2019

References

Samuel M. Selby, editor, CRC Basic Mathematical Tables, CRC Press, 1970, p. 218.

Links

Ivan Panchenko, Table of n, a(n) for n = 1..1000

Hideyuki Ohtsuka, Problem 11853, The American Mathematical Monthly, Vol. 122, No. 7 (2015), p. 700; A Hyperbolic Sine Series, Solutions to Problem 11853 by Tewodros Amdeberhan and Rituraj Nandan, ibid., Vol. 124, No. 5 (2017), p. 469.

Allen Stenger, Experimental math for Math Monthly problems, The American Mathematical Monthly, Vol. 124, No. 2 (2017), pp. 116-131; alternative link.

Eric Weisstein's World of Mathematics, Hyperbolic Cotangent.

Eric Weisstein's World of Mathematics, Hyperbolic Functions.

Index entries for transcendental numbers

Formula

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

Equals 1 + BesselI(3/2, 1)/BesselI(1/2, 1). - Terry D. Grant, Jun 18 2018

Equals 1 + Sum_{k>=1} csch(2^k) (Ohtsuka, 2015; Stenger, 2017). - Amiram Eldar, Oct 04 2021

Example

1.31303528549933130363616124693...

Mathematica

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 *)

Prog

(PARI) 1/tanh(1)

Crossrefs

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.

Sequence in context: A253626 A112298 A011430 * A349004 A127549 A191009

Adjacent sequences: A073744 A073745 A073746 * A073748 A073749 A073750

Keyword

cons,nonn

Author

Rick L. Shepherd, Aug 07 2002

Status

approved