A073745 -id:A073745 - cp's OEIS frontend

A073743 Decimal expansion of cosh(1).

1, 5, 4, 3, 0, 8, 0, 6, 3, 4, 8, 1, 5, 2, 4, 3, 7, 7, 8, 4, 7, 7, 9, 0, 5, 6, 2, 0, 7, 5, 7, 0, 6, 1, 6, 8, 2, 6, 0, 1, 5, 2, 9, 1, 1, 2, 3, 6, 5, 8, 6, 3, 7, 0, 4, 7, 3, 7, 4, 0, 2, 2, 1, 4, 7, 1, 0, 7, 6, 9, 0, 6, 3, 0, 4, 9, 2, 2, 3, 6, 9, 8, 9, 6, 4, 2, 6, 4, 7, 2, 6, 4, 3, 5, 5, 4, 3, 0, 3, 5, 5, 8, 7, 0, 4

Offset: 1

Rick L. Shepherd, Aug 07 2002

Also decimal expansion of cos(i). - N. J. A. Sloane, Feb 12 2010
cosh(x) = (e^x + e^(-x))/2.
Equals Sum_{n>=0} 1/A010050(n). See Gradsteyn-Ryzhik (0.245.5). - R. J. Mathar, Oct 27 2012
By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 14 2019

			1.54308063481524377847790562075...

S. Selby, editor, CRC Basic Mathematical Tables, CRC Press, 1970, p. 218.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 2, equation 2:5:6 at page 20.

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Hyperbolic Cosine.
Eric Weisstein's World of Mathematics, Hyperbolic Functions.
Eric Weisstein's World of Mathematics, Factorial Sums.
Index entries for transcendental numbers.

Cf. A068118 (continued fraction), A073742, A073744, A073745, A073746, A073747, A049470, A137204.

Digits:=100: evalf(cosh(1)); # Wesley Ivan Hurt, Nov 18 2014

RealDigits[Cosh[1],10,120][[1]] (* Harvey P. Dale, Aug 03 2014 *)

PARI
```
cosh(1)
```

Continued fraction representation: cosh(1) = 1 + 1/(2 - 2/(13 - 12/(31 - ... - (2*n - 4)*(2*n - 5)/((4*n^2 - 10*n + 7) - ... )))). See A051396 for proof. Cf. A049470 (cos(1)) and A073742 (sinh(1)). - Peter Bala, Sep 05 2016
Equals Product_{k>=0} 1 + 4/((2*k+1)*Pi)^2. - Amiram Eldar, Jul 16 2020
Equals 1/A073746 = A137204/2. - Hugo Pfoertner, Dec 27 2024

A073742 Decimal expansion of sinh(1).

Original entry on oeis.org

1, 1, 7, 5, 2, 0, 1, 1, 9, 3, 6, 4, 3, 8, 0, 1, 4, 5, 6, 8, 8, 2, 3, 8, 1, 8, 5, 0, 5, 9, 5, 6, 0, 0, 8, 1, 5, 1, 5, 5, 7, 1, 7, 9, 8, 1, 3, 3, 4, 0, 9, 5, 8, 7, 0, 2, 2, 9, 5, 6, 5, 4, 1, 3, 0, 1, 3, 3, 0, 7, 5, 6, 7, 3, 0, 4, 3, 2, 3, 8, 9, 5, 6, 0, 7, 1, 1, 7, 4, 5, 2, 0, 8, 9, 6, 2, 3, 3, 9, 1, 8, 4, 0, 4, 1

Offset: 1

Rick L. Shepherd, Aug 07 2002

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

Decimal expansion of u > 0 such that 1 = arclength on the hyperbola y^2 - x^2 = 1 from (0,0) to (u,y(u)). - Clark Kimberling, Jul 04 2020

			1.17520119364380145688238185059...

S. Selby, editor, CRC Basic Mathematical Tables, CRC Press, 1970, p. 218.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 2, equation 2:5:7 at page 20.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Eric Weisstein's World of Mathematics, Hyperbolic Sine.
Eric Weisstein's World of Mathematics, Hyperbolic Functions.
Eric Weisstein's World of Mathematics, Factorial Sums.
Index entries for transcendental numbers.

Cf. A068139 (continued fraction), A073743, A073744, A073745, A073746, A073747, A049469, A049470, A174548.

First@ RealDigits@ N[Sinh@ 1, 120] (* Michael De Vlieger, Sep 04 2016 *)

PARI
```
sinh(1)
```

Equals (e - e^(-1))/2.
Equals sin(i)/i. - N. J. A. Sloane, Feb 12 2010
Equals Sum_{n>=0} 1/A009445(n). See Gradsteyn-Ryzhik (0.245.6.) - R. J. Mathar, Oct 27 2012
Continued fraction representation: sinh(1) = 1 + 1/(6 - 6/(21 - 20/(43 - 42/(73 - ... - (2*n - 1)*(2*n - 2)/((2*n*(2*n + 1) + 1) - ... ))))). See A051397 for proof. Cf. A049469. - Peter Bala, Sep 02 2016
Equals Product_{k>=1} 1 + 1/(k * Pi)^2. - Amiram Eldar, Jul 16 2020
Equals 1/A073745 = A174548/2. - Hugo Pfoertner, Dec 27 2024

A073747 Decimal expansion of coth(1).

Original entry on oeis.org

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

Rick L. Shepherd, Aug 07 2002

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

			1.31303528549933130363616124693...

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

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

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.

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 *)

PARI
```
1/tanh(1)
```

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

A073744 Decimal expansion of tanh(1).

Original entry on oeis.org

7, 6, 1, 5, 9, 4, 1, 5, 5, 9, 5, 5, 7, 6, 4, 8, 8, 8, 1, 1, 9, 4, 5, 8, 2, 8, 2, 6, 0, 4, 7, 9, 3, 5, 9, 0, 4, 1, 2, 7, 6, 8, 5, 9, 7, 2, 5, 7, 9, 3, 6, 5, 5, 1, 5, 9, 6, 8, 1, 0, 5, 0, 0, 1, 2, 1, 9, 5, 3, 2, 4, 4, 5, 7, 6, 6, 3, 8, 4, 8, 3, 4, 5, 8, 9, 4, 7, 5, 2, 1, 6, 7, 3, 6, 7, 6, 7, 1, 4, 4, 2, 1, 9, 0

Offset: 0

Rick L. Shepherd, Aug 07 2002

Also decimal expansion of tan(i)/i. - N. J. A. Sloane, Feb 12 2010
tanh(x) = (e^x - e^(-x)) / (e^x + e^(-x)).
By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 14 2019

			0.76159415595576488811945828260...

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

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Hyperbolic Tangent
Eric Weisstein's World of Mathematics, Hyperbolic Functions
Index entries for transcendental numbers

Cf. A004273 (continued fraction), A073747 (coth(1)=1/A073744), A073742 (sinh(1)), A073743 (cosh(1)), A073745 (csch(1)), A073746 (sech(1)).

Cf. A027641, A027642, A349003.

RealDigits[Tanh[1], 10, 100][[1]] (* Amiram Eldar, Aug 19 2020 *)

PARI
```
tanh(1)
```

Equals Sum_{k>=1} bernoulli(2*k)*2^(2*k)*(2^(2*k)-1)/(2*k)!, where bernoulli(k) = A027641(k)/A027642(k) is the k-th Bernoulli number. - Amiram Eldar, Aug 19 2020
Equal to the continued fraction [0;1,3,5,...,2n-1,...]. - Thomas Ordowski, Oct 22 2024
Equals 1-A349003. - Hugo Pfoertner, Oct 22 2024

A073746 Decimal expansion of sech(1).

Original entry on oeis.org

6, 4, 8, 0, 5, 4, 2, 7, 3, 6, 6, 3, 8, 8, 5, 3, 9, 9, 5, 7, 4, 9, 7, 7, 3, 5, 3, 2, 2, 6, 1, 5, 0, 3, 2, 3, 1, 0, 8, 4, 8, 9, 3, 1, 2, 0, 7, 1, 9, 4, 2, 0, 2, 3, 0, 3, 7, 8, 6, 5, 3, 3, 7, 3, 1, 8, 7, 1, 7, 5, 9, 5, 6, 4, 6, 7, 1, 2, 8, 3, 0, 2, 8, 0, 8, 5, 4, 7, 8, 5, 3, 0, 7, 8, 9, 2, 8, 9, 2, 3, 8, 4, 8, 4

Offset: 0

Rick L. Shepherd, Aug 07 2002

sech(x) = 2/(e^x + e^(-x)).

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

			0.64805427366388539957497735322...

Samuel M. Selby (ed.), CRC Basic Mathematical Tables, CRC Press, 1970, p. 218.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Hyperbolic Secant.
Eric Weisstein's World of Mathematics, Hyperbolic Functions.
Index entries for transcendental numbers

Cf. A068118 (continued fraction), A073743 (cosh(1)=1/A073746), A073742 (sinh(1)), A073744 (tanh(1)), A073745 (csch(1)), A073747 (coth(1)), A122045.

RealDigits[Sech[1], 10, 100][[1]] (* Amiram Eldar, May 15 2021 *)

PARI
```
1/cosh(1)
```

Equals Sum_{k>=0} E(2*k) / (2*k)!, where E(k) is the k-th Euler number (A122045). - Amiram Eldar, May 15 2021

A068139 Continued fraction expansion for sinh(1).

Original entry on oeis.org

1, 5, 1, 2, 2, 2, 1, 2, 7, 5, 1, 1, 1, 2, 2, 19, 1, 2, 1, 7, 1, 1, 9, 1, 3, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 4, 5, 3, 5, 1, 3, 1, 1, 1, 2, 7, 1, 9, 1, 1, 2, 1, 21, 1, 18, 1, 2, 734, 1, 2, 1, 84, 2, 2, 1, 1, 2, 10, 1, 3, 7, 1, 16, 1, 2, 4, 56, 1, 13, 16, 208

Offset: 0

Benoit Cloitre, Mar 13 2002

If an extra zero is added to the beginning of this sequence, continued fraction for csch(1) = 1/sinh(1). - Rick L. Shepherd, Aug 07 2002

Robert Israel, Table of n, a(n) for n = 0..9999

Cf. A068118, A073742 (decimal expansion), A073745 (decimal expansion of csch(1)).

Cf. A078980, A078981 (convergents).

numtheory:-cfrac(sinh(1),100,quotients)[1..-2]; # Robert Israel, Nov 14 2020

ContinuedFraction[Sinh@ 1, 82] (* Michael De Vlieger, Sep 04 2016 *)

contfrac(sinh(1)) \\ Michel Marcus, Sep 04 2016

Offset changed by Andrew Howroyd, Aug 05 2024

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A073743 Decimal expansion of cosh(1).

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A073742 Decimal expansion of sinh(1).

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A073747 Decimal expansion of coth(1).

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A073744 Decimal expansion of tanh(1).

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A073746 Decimal expansion of sech(1).

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A068139 Continued fraction expansion for sinh(1).

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