A073742 -id:A073742 - 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

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

A143819 Decimal expansion of Sum_{k>=0} 1/(3*k)!.

Original entry on oeis.org

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

Offset: 1

Peter Bala, Sep 03 2008

Previous name was: Decimal expansion of the constant 1 + 1/3! + 1/6! + 1/9! + ... = 1.16805 83133 75918 ... .
Define a sequence R(n) of real numbers by R(n) := Sum_{k>=0} (3*k)^n/(3*k)! for n = 0,1,2,... . This constant is R(0); the decimal expansions of R(2) - R(1) = 1/1! + 1/4! + 1/7! and R(1) = 1/2! + 1/5! + 1/8! + ... may be found in A143820 and A143821. It is easy to verify that the sequence R(n) satisfies the recurrence relation u(n+3) = 3*u(n+2) - 2*u(n+1) + Sum_{i=0..n} binomial(n,i) * 3^(n-i)*u(i). Hence R(n) is an integral linear combination of R(0), R(1) and R(2) and so also an integral linear combination of R(0), R(1) and R(2) - R(1). Some examples are given below.
Bowman and Mc Laughlin (Corollary 10 with m = -1) give a continued fraction expansion for this constant and deduce the constant is irrational. - Peter Bala, Apr 17 2017

			1.168058313375918525516256929611144747716933295113292516385891232685...
R(n) as a linear combination of R(0), R(1) and R(2) - R(1).
=======================================
  R(n)  |     R(0)     R(1)   R(2)-R(1)
=======================================
  R(3)  |       1        1        3
  R(4)  |       6        2        7
  R(5)  |      25       11       16
  R(6)  |      91       66       46
  R(7)  |     322      352      203
  R(8)  |    1232     1730     1178
  R(9)  |    5672     8233     7242
  R(10) |   32202    39987    43786
  ...
The column entries are from A143815, A143816 and A143817.

D. Bowman and J. Mc Laughlin, Polynomial continued fractions, arXiv:1812.08251 [math.NT], 2018; Acta Arith. 103 (2002), no. 4, 329-342.
Michael Penn, Two sum identities, YouTube video, 2020.
Michael I. Shamos, A catalog of the real numbers, (2011). See p. 209.

Cf. A073742, A073743, A143815, A143816, A143817, A143818, A143820, A143821.

Cf. A001113 (Sum 1/k!), A073743 (Sum 1/(2k)!), this sequence (Sum 1/(3k)!), A332890 (Sum 1/(4k)!), A269296 (Sum 1/(5k)!), A332892 (Sum 1/(6k)!), A346441.

RealDigits[ N[ 1/3*(2*Cos[Sqrt[3]/2]/Sqrt[E] + E), 105]][[1]] (* Jean-François Alcover, Nov 08 2012 *)
With[{nn=120},RealDigits[N[Total[Table[1/(3n)!,{n,nn}]]+1,nn],10,nn][[1]]] (* Harvey P. Dale, Apr 20 2013 *)

suminf(k=0, 1/(3*k)!) \\ Michel Marcus, Feb 21 2016

Equals (exp(1) + exp(w) + exp(w^2))/3, where w = exp(2*Pi*i/3).
A143819 + A143820 + A143821 = exp(1).
Equals 1/3 * (e + 2 * cos(sqrt(3)/2) / sqrt(e)). - Bernard Schott, Mar 01 2020
Sum_{k>=0} (-1)^k / (3*k)! = (exp(-1) + 2*exp(1/2)*cos(sqrt(3)/2))/ 3 = 0.83471946857721... - Vaclav Kotesovec, Mar 02 2020
Continued fraction: 1 + 1/(6 - 6/(121 - 120/(505 - ... - P(n-1)/((P(n) + 1) - ... )))), where P(n) = (3*n )*(3*n - 1)*(3*n - 2) for n >= 1. Cf. A346441. - Peter Bala, Feb 22 2024

Offset corrected by R. J. Mathar, Feb 05 2009

New name from Bernard Schott, Mar 02 2020

A051397 a(n) = (2n-2)(2n-1)a(n-1)+1.

Original entry on oeis.org

0, 1, 7, 141, 5923, 426457, 46910271, 7318002277, 1536780478171, 418004290062513, 142957467201379447, 60042136224579367741, 30381320929637160076947, 18228792557782296046168201, 12796612375563171824410077103, 10390849248957295521420982607637

Offset: 0

Aleksandar Petojevic

Harvey P. Dale, Table of n, a(n) for n = 0..225
Romeo Mestrovic, Variations of Kurepa's left factorial hypothesis, arXiv preprint arXiv:1312.7037 [math.NT], 2013.
Romeo Mestrovic, The Kurepa-Vandermonde matrices arising from Kurepa's left factorial hypothesis, Filomat 29:10 (2015), 2207-2215; DOI 10.2298/FIL1510207M.
Aleksandar Petojevic, On Kurepa's Hypothesis for the Left Factorial, FILOMAT (Nis), 12:1 (1998), p. 29-37.

Bisection of abs(A009628). Also bisection of A087208 and of A186763. Cf. A073742, A074790, A275651.

nxt[{n_,a_}]:={n+1,(2(n+1)-2)(2(n+1)-1)a+1}; Transpose[NestList[nxt,{0,0},20]][[2]] (* Harvey P. Dale, Jun 13 2016 *)

a(n) = Sum_{k=0..n-1} (2*n-1)!/(2*k+1)!. a(n) = floor((2*n-1)!*sinh(1)). - Vladeta Jovovic, Aug 10 2002
Conjecture: a(n) +(-4*n^2+6*n-3)*a(n-1) +2*(2*n-3)*(n-2)*a(n-2)=0. - R. J. Mathar, Jan 31 2014
From Peter Bala, Sep 02 2016: (Start)
G.f. sinh(x)/(1 - x^2) = x + 7*x^3/3! + 141*x^5/5! + 5923*x^7/7! + ....
Mathar's conjectured recurrence a(n) = (4*n^2 - 6*n + 3)*a(n-1) - (2*n - 3)*(2*n - 4)*a(n-2) follows easily from the defining recurrence. The sequence b(n) := (2*n - 1)! also satisfies Mathar's recurrence but with b(1) = 1, b(2) = 6. This leads to the continued fraction representation a(n) = (2*n - 1)!*(1 + 1/(6 - 6/(21 - 20/(43 - ... - (2*n - 3)*(2*n - 4)/(4*n^2 - 6*n + 3) )))) for n >= 3. Taking the limit gives the continued fraction representation sinh(1) = A073742 = 1 + 1/(6 - 6/(21 - 20/(43 - ... - (2*n - 3)*(2*n - 4)/((4*n^2 - 6*n + 3) - ... )))). (End)

A143820 Decimal expansion of the constant 1/1! + 1/4! + 1/7! + ...

Original entry on oeis.org

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

Offset: 1

Peter Bala, Sep 03 2008

Define a sequence R(n) of real numbers by R(n) := Sum_{k >= 0} (3*k)^n/(3*k)! for n = 0,1,2,... . This constant is R(2) - R(1); the decimal expansions of R(0) = 1 + 1/3! + 1/6! + ... and R(1) = 1/2! + 1/5! + 1/8! + ... may be found in A143819 and A143821. It is easy to verify that the sequence R(n) satisfies the recurrence relation u(n+3) = 3*u(n+2) - 2*u(n+1) + Sum_{i = 0..n} binomial(n,i) * 3^(n-i)*u(i). Hence R(n) is an integral linear combination of R(0), R(1) and R(2) and so also an integral linear combination of R(0), R(1) and R(2) - R(1).
R(n) as a linear combination of R(0), R(1) and R(2) - R(1).
========================================
        |       linear combination of
  R(n)  |     R(0)     R(1)  R(2) - R(1)
========================================
  R(3)  |       1        1        3
  R(4)  |       6        2        7
  R(5)  |      25       11       16
  R(6)  |      91       66       46
  R(7)  |     322      352      203
  R(8)  |    1232     1730     1178
  R(9)  |    5672     8233     7242
  R(10) |   32202    39987    43786
  ...
The column entries are from A143815, A143816 and A143817.
The Abraham Ungar 1982 article defines H_{n,r}(z) = Sum_{k>=0} z^(nk+r)/(nk+r)! as equation (1). The constant is H_{3,1}(1). In equation (13) H_{3,1}(x) = (exp(x) + 2 * exp(-x/2) * cos(sqrt(3)/2*x - 2*Pi/3))/3. In equation (12) the expression H_{3,1}(x) = (e^x + q_2 e^{q_1 x} + q_1 e^{q_2 x})/3 where q_1 = (-1 + I sqrt(3))/2 and q_2 = (-1 - I sqrt(3))/2 is given for H_{3,2}(x) instead. - Michael Somos, Nov 01 2024

			1.041865355098909...

Michael I. Shamos, A catalog of the real numbers, (2011). See p. 76.
Abraham Ungar, Generalized Hyperbolic Functions, The American Mathematical Monthly, Volume 89, 1982, 688-691.

Cf. A073742, A073743, A143815, A143816, A143817, A143818, A143819, A143821, A346441.

Digits:=101: evalf(sum(1/(3*n+1)!, n=0..infinity)); # Michal Paulovic, Aug 20 2023

RealDigits[ N[ (-Cos[Sqrt[3]/2] + E^(3/2) + Sqrt[3]*Sin[Sqrt[3]/2])/(3*Sqrt[E]), 105]][[1]] (* Jean-François Alcover, Nov 08 2012 *)

suminf(n=0,1/(3*n+1)!) \\ Michel Marcus, Aug 20 2023

Equals (exp(1) + w^2*exp(w) + w*exp(w^2))/3, where w = exp(2*Pi*i/3).
A143819 + A143820 + A143821 = exp(1).
Equals Sum_{n>=0} 1/(3*n+1)!. - Michal Paulovic, Aug 20 2023
Continued fraction: 1 + 1/(24 - 24/(211 - 210/(721 - ... - P(n-1)/((P(n) + 1) - ... )))), where P(n) = (3*n - 1)*(3*n)*(3*n + 1) for n >= 1. Cf. A346441. - Peter Bala, Feb 22 2024
Equals (exp(1) + 2*exp(-1/2)*cos(sqrt(3)/2-2*Pi/3))/3. [Ungar, p.690] - Michael Somos, Nov 01 2024

Offset corrected by R. J. Mathar, Feb 05 2009

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

A143821 Decimal expansion of the constant 1/2! + 1/5! + 1/8! + ... = 0.50835 81599 84216 ... .

Original entry on oeis.org

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

Offset: 0

Peter Bala, Sep 03 2008

Define a sequence of real numbers R(n) by R(n) := Sum_{k >= 0} (3*k)^n/(3*k)! for n = 0,1,2... . This constant is R(1); the decimal expansions of R(0) = 1 + 1/3!+ 1/6! + 1/9! + ... and R(2) - R(1) = 1/1! + 1/4! + 1/7! + ... may be found in A143819 and A143820. It is easy to verify that the sequence R(n) satisfies the recurrence relation u(n+3) = 3*u(n+2) - 2*u(n+1) + Sum_{i = 0..n} binomial(n,i) *3^(n-i)*u(i). Hence R(n) is an integral linear combination of R(0), R(1) and R(2) and so also an integral linear combination of R(0), R(1) and R(2) - R(1). Some examples are given below.

			R(n) as a linear combination of R(0), R(1) and R(2) - R(1).
=======================================
..R(n)..|.....R(0).....R(1)...R(2)-R(1)
=======================================
..R(3)..|.......1........1........3....
..R(4)..|.......6........2........7....
..R(5)..|......25.......11.......16....
..R(6)..|......91.......66.......46....
..R(7)..|.....322......352......203....
..R(8)..|....1232.....1730.....1178....
..R(9)..|....5672.....8233.....7242....
..R(10).|...32202....39987....43786....
...
The column entries are from A143815, A143816 and A143817.

Michael I. Shamos, A catalog of the real numbers, (2011). See p. 460.

Cf. A073742, A073743, A143815, A143816, A143817, A143818, A143819, A143820, A346441.

RealDigits[ N[ -((Cos[Sqrt[3]/2] - E^(3/2) + Sqrt[3]*Sin[Sqrt[3]/2])/(3*Sqrt[E])), 105]][[1]] (* Jean-François Alcover, Nov 08 2012 *)

Constant = (exp(1) + w*exp(w) + w^2*exp(w^2))/3, where w = exp(2*Pi*i/3). A143819 + A143820 + A143821 = exp(1).

Continued fraction: 1/(2 - 2/(61 - 60/(337 - 336/(991 - ... - P(n-1)/((P(n) + 1) - ... ))))), where P(n) = (3*n)*(3*n + 1)*(3*n + 2) for n >= 1. Cf. A346441. - Peter Bala, Feb 22 2024

Offset corrected by R. J. Mathar, Feb 05 2009

A068377 Engel expansion of sinh(1).

Original entry on oeis.org

1, 6, 20, 42, 72, 110, 156, 210, 272, 342, 420, 506, 600, 702, 812, 930, 1056, 1190, 1332, 1482, 1640, 1806, 1980, 2162, 2352, 2550, 2756, 2970, 3192, 3422, 3660, 3906, 4160, 4422, 4692, 4970, 5256, 5550, 5852, 6162, 6480, 6806, 7140, 7482, 7832, 8190

Offset: 1

Benoit Cloitre, Mar 03 2002

This sequence is also the Pierce expansion of sin(1). - G. C. Greubel, Nov 14 2016

Simon Plouffe, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Engel Expansion.
Eric Weisstein's World of Mathematics, Hyperbolic Sine.
Eric Weisstein's World of Mathematics, Pierce Expansion.
Wikipedia, Engel Expansion.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A002943, A006784, A068379, A068380, A073742 (sinh(1)).

Join[{1}, Table[(2 n - 2) (2 n - 1), {n, 2, 50}]] (* Bruno Berselli, Aug 04 2015 *)
LinearRecurrence[{3,-3,1}, {1,6,20,42}, 25] (* G. C. Greubel, Oct 27 2016; a(1)=1 by Georg Fischer, Apr 02 2019*)
Rest@ CoefficientList[Series[x (1 + 3 x + 5 x^2 - x^3)/(1 - x)^3, {x, 0, 46}], x] (* Michael De Vlieger, Oct 28 2016 *)
PierceExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@ NestList[{Floor[1/Expand[1 - #[[1]] #[[2]]]], Expand[1 - #[[1]] #[[2]]]} &, {Floor[1/(A - Floor[A])], A - Floor[A]}, n - 1]]; PierceExp[N[Sin[1] , 7!], 50] (* G. C. Greubel, Nov 14 2016 *)

A068377(n)=(n+n--)*n*2+!n \\ M. F. Hasler, Jul 19 2015

A068377 = lambda n: rising_factorial(n*2,2) if n>0 else 1
print([A068377(n) for n in (0..45)]) # Peter Luschny, Aug 04 2015

a(n) = (2*n-2)*(2*n-1) = A002943(n-1) = 2*A000217(2n-2) for n>1. [Corrected and extended by M. F. Hasler, Jul 19 2015]
From Colin Barker, Apr 13 2012: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>4.
G.f.: x*(1 + 3*x + 5*x^2 - x^3)/(1-x)^3. (End)
E.g.f.: -2 + x + 2*(1 - x + 2*x^2)*exp(x). - G. C. Greubel, Oct 27 2016
From Amiram Eldar, May 05 2025: (Start)
Sum_{n>=1} 1/a(n) = 2 - log(2).
Sum_{n>=1} (-1)^(n+1)/a(n) = 2 - Pi/4 - log(2)/2. (End)

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

cp's OEIS Frontend

A073743 Decimal expansion of cosh(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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A143819 Decimal expansion of Sum_{k>=0} 1/(3*k)!.

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A051397 a(n) = (2*n-2)*(2*n-1)*a(n-1)+1.

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A143820 Decimal expansion of the constant 1/1! + 1/4! + 1/7! + ...

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A073746 Decimal expansion of sech(1).

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A051397 a(n) = (2n-2)(2n-1)a(n-1)+1.