A068139 -id:A068139 - cp's OEIS frontend

A073742 Decimal expansion of sinh(1).

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

A073745 Decimal expansion of csch(1).

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Aug 07 2002

csch(x) = 2/(e^x - e^(-x)).

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

			0.85091812823932154513384276328...

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 Cosecant.
Eric Weisstein's World of Mathematics, Hyperbolic Functions.
Index entries for transcendental numbers

Cf. A068139 (continued fraction), A073742 (sinh(1)=1/A073745), A073743 (cosh(1)), A073744 (tanh(1)), A073746 (sech(1)), A073747 (coth(1)).

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

PARI
```
1/sinh(1)
```

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

A068118 Continued fraction expansion for cosh(1).

Original entry on oeis.org

1, 1, 1, 5, 3, 3, 2, 1, 21, 1, 1, 1, 4, 2, 1, 48, 71, 7, 1, 1, 9, 1, 2, 1, 11, 1, 5, 1, 2, 3, 2, 2, 2, 1, 1, 2, 10, 1, 1, 5, 26, 1, 25, 1, 2, 1, 5, 1, 2, 2, 7, 1, 1, 8, 8, 2, 1, 7, 2, 1, 9, 1, 3, 1, 4, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 5, 1, 1, 3, 1, 1, 1, 3, 7, 183

Offset: 0

Benoit Cloitre, Mar 13 2002

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

Cf. A068139, A073743 (decimal expansion), A073746 (decimal expansion of sech(1)).

Cf. A078982, A078983 (convergents).

ContinuedFraction[Cosh[1],90] (* Harvey P. Dale, Apr 29 2013 *)

Offset changed by Andrew Howroyd, Aug 05 2024

A078980 Numerators of continued fraction convergents to sinh(1).

Original entry on oeis.org

1, 6, 7, 20, 47, 114, 161, 436, 3213, 16501, 19714, 36215, 55929, 148073, 352075, 6837498, 7189573, 21216644, 28406217, 220060163, 248466380, 468526543, 4465205267, 4933731810, 19266400697, 24200132507, 43466533204, 111133198915

Offset: 0

Benoit Cloitre, Dec 20 2002

Cf. A068139 (continued fraction), A073742 (decimal expansion), A078981 (denominators).

Numerator[Convergents[Sinh[1],30]] (* Harvey P. Dale, May 09 2013 *)

a(n)=component(component(contfracpnqn(contfrac(sinh(1),n+1)),1),1)

Offset changed by Andrew Howroyd, Aug 05 2024

A078981 Denominators of continued fraction convergents to sinh(1).

Original entry on oeis.org

1, 5, 6, 17, 40, 97, 137, 371, 2734, 14041, 16775, 30816, 47591, 125998, 299587, 5818151, 6117738, 18053627, 24171365, 187253182, 211424547, 398677729, 3799524108, 4198201837, 16394129619, 20592331456, 36986461075, 94565253606

Offset: 0

Benoit Cloitre, Dec 20 2002

Cf. A068139 (continued fraction), A073742 (decimal expansion), A078980 (numerators).

Convergents[Sinh[1],30]//Denominator (* Harvey P. Dale, Apr 17 2022 *)

a(n)=component(component(contfracpnqn(contfrac(sinh(1),n+1)),1),2)

Offset changed by Andrew Howroyd, Aug 05 2024

cp's OEIS Frontend

A073742 Decimal expansion of sinh(1).

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A073745 Decimal expansion of csch(1).

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A068118 Continued fraction expansion for cosh(1).

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A078980 Numerators of continued fraction convergents to sinh(1).

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A078981 Denominators of continued fraction convergents to sinh(1).

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Extensions