A073747 -id:A073747 - cp's OEIS frontend

A109237 a(n) = floor(n*coth(1)).

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 24, 26, 27, 28, 30, 31, 32, 34, 35, 36, 38, 39, 40, 42, 43, 44, 45, 47, 48, 49, 51, 52, 53, 55, 56, 57, 59, 60, 61, 63, 64, 65, 66, 68, 69, 70, 72, 73, 74, 76, 77, 78, 80, 81, 82, 84, 85, 86

Offset: 1

Reinhard Zumkeller, Jun 23 2005

nonn

Beatty sequence for coth(1) = (e^2+1)/(e^2-1) = 1.31303... = A073747; complement of A109238.
From Reinhard Zumkeller, Aug 11 2009: (Start)
a(n) = A164087(n) for n <= 20;
a(n) = A109239(A109238(n)) and A109239(a(n)) = A109238(n). (End)

Eric Weisstein's World of Mathematics, Hyperbolic Cotangent
Eric Weisstein's World of Mathematics, Beatty Sequence
Index entries for sequences related to Beatty sequences

Cf. A109238, A109239, A109231, A109234.

Cf. A073747.

Typo in comment corrected by Reinhard Zumkeller, Aug 10 2009

A298241 Decimal expansion of BesselI(1,2/3)/BesselI(0,2/3).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Jan 15 2018

			0.3160892412682211840956016917105181147668629270070418207...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Continued Fraction Constants
Index entries for sequences related to Bessel functions or polynomials

Cf. A008585, A052119, A073744, A073747, A073821, A296168, A298242, A298243.

RealDigits[BesselI[1, 2/3]/BesselI[0, 2/3], 10, 110] [[1]]
RealDigits[Hypergeometric0F1[2, (2/3)^2/4] /(3 Gamma[2] Hypergeometric0F1[1, (2/3)^2/4]), 10, 110][[1]]

besseli(1,2/3)/besseli(0,2/3) \\ Michel Marcus, Jul 03 2018

Equals 1/(3 + 1/(6 + 1/(9 + 1/(12 + 1/(15 + 1/(18 + ...)))))).

A298243 Decimal expansion of BesselI(1,2/5)/BesselI(0,2/5).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Jan 15 2018

			0.1961038122179955134083610646268785173725058094464270021...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Continued Fraction Constants
Index entries for sequences related to Bessel functions or polynomials

Cf. A008587, A052119, A073744, A073747, A073821, A296168, A298241, A298242.

RealDigits[BesselI[1, 2/5]/BesselI[0, 2/5], 10, 110] [[1]]
RealDigits[Hypergeometric0F1[2, (2/5)^2/4]/(5 Gamma[2] Hypergeometric0F1[1, (2/5)^2/4]), 10, 110][[1]]

besseli(1,2/5)/besseli(0,2/5) \\ Michel Marcus, Jul 03 2018

Equals 1/(5 + 1/(10 + 1/(15 + 1/(20 + 1/(25 + 1/(30 + ...)))))).

A036244 Denominator of continued fraction given by C(n) = [ 1; 3, 5, 7, ...(2n-1)].

Original entry on oeis.org

1, 3, 16, 115, 1051, 11676, 152839, 2304261, 39325276, 749484505, 15778499881, 363654981768, 9107153044081, 246256787171955, 7150553981030776, 221913430199126011, 7330293750552189139, 256782194699525745876, 9508271497633004786551, 371079370602386712421365

Offset: 1

Jeff Burch

Denominators of convergents to coth(1) = 1.313035... = A073747.

Convergents: 1/1, 4/3, 21/16, 151/115, ... - Michael Somos, Sep 27 2017

			G.f. = x + 3*x^2 + 16*x^3 + 115*x^4 + 1051*x^5 + 11676*x^6 + 152839*x^7 + ...

Robert Israel, Table of n, a(n) for n = 1..369

Cf. A001040, A001053, A073747.

Numerators are sequence A025164. A058798.

I:=[1,3]; [n le 2 select I[n] else (2*n-1)*Self(n-1)+Self(n-2): n in [1..30]]; // Vincenzo Librandi, Apr 19 2015

seq(denom(numtheory:-cfrac([seq(2*i-1,i=1..n)])),n=1..50); # Robert Israel, Apr 19 2015

Rest[CoefficientList[Series[(E^(1-(1-2*x)^(1/2))/2 - E^(-1+(1-2*x)^(1/2))/2) / (1-2*x)^(1/2), {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec, Oct 05 2013 *)
a[ n_ ] := a[n] =a[n-2]+(2 n-1) a[n-1]; a[0] := 0; a[1] := 1.  RecurrenceTable[{a[0]==0, a[1]==1, a[n]==a[n-2]+(2n-1)a[n-1]}, a, {n, 20}] (* G. C. Greubel,Apr 23 2015 *)
a[ n_] := (BesselK[ 1/2, 1] BesselI[ n + 1/2, -1] - BesselI[ 1/2, -1] BesselK[n + 1/2, 1])  I // FunctionExpand // Simplify; (* Michael Somos, Sep 27 2017 *)
Table[FromContinuedFraction[Range[1,2n+1,2]],{n,0,20}]//Denominator (* Harvey P. Dale, May 06 2018 *)
Convergents[Coth[1], 20] // Denominator (* Jean-François Alcover, Jun 15 2019 *)

def A036244(n):
    if n == 1: return 1
    return 2^n*gamma(n+1/2)*hypergeometric([1/2-n/2, 1-n/2], [3/2, 1/2-n, 1-n], 1)/sqrt(pi)
[round(A036244(n).n(100)) for n in (1..20)] # Peter Luschny, Sep 11 2014

a(n) = a(n-1)*(2*n-1) + a(n-2); a(0) = 0, a(1) = 1.
E.g.f.: sinh(1-(1-2*x)^(1/2))/(1-2*x)^(1/2). - Vladeta Jovovic, Jan 30 2004
E.g.f.: cosh(1-(1-2*x)^(1/2))/(1-2*x) + sinh(1-(1-2*x)^(1/2))/((1-2*x)^(3/2)).
E.g.f. G(0)/(1-2*x) where G(k)= 1 + 2*x/((2*k+1)*(1-2*x+sqrt(1-2*x))+(2*k+1)*(4*x^2-2*x)/(-1+2*x+sqrt(1-2*x) + (2*k+2)/G(k+1))); (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Jul 01 2012
a(n) = Sum_{k=0..floor((n-1)/2)} 2^(n-2*k-1)*(n-2*k-1)!*binomial(n-k-1,k)*binomial(n-k-1/2,k+1/2). Cf. A058798. - Peter Bala, Aug 01 2013
a(n) ~ (exp(2)-1)*2^(n-1/2)*n^n/exp(n+1). - Vaclav Kotesovec, Oct 05 2013
a(n) = A001147(n)*hypergeometric([1/2-n/2, 1-n/2], [3/2, 1/2-n, 1-n], 1) for n >= 2. - Peter Luschny, Sep 11 2014
a(n) = i*(BesselK[1/2,1]*BesselI[n+1/2,-1] - BesselI[1/2,-1]*BesselK[n+1/2,1]) for n>=0 (where a(0) = 0). - G. C. Greubel, Apr 18 2015
a(n) = A025164(-1-n) for all n in Z. - Michael Somos, Sep 27 2017

More terms from Larry Reeves (larryr(AT)acm.org), May 15 2001
More terms from Benoit Cloitre, Dec 20 2002
More terms from Vladeta Jovovic, Jan 30 2004

A349004 Decimal expansion of lim_{n->infinity} B(2n, n)/n^(2n), where B(n, x) is the n-th Bernoulli polynomial.

Original entry on oeis.org

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, 4, 7

Offset: 0

Vaclav Kotesovec, Nov 05 2021

Asymptotic expansion: B(2*n,n) / n^(2*n) ~ c0 + c1/n + c2/n^2 + ..., where
c0 = A349004
c1 = -0.11332842437985451266688985513574347679739396134203607414578687657...
c2 = -0.02939332883129837328682967905833985820907100422772261310141242364...
In general, for k>=1, B(k*n,n) / n^(k*n) ~ k/(exp(k) - 1).

			0.313035285499331303636161246930847832912013941240452655543152967567084...

William Bell, Problem 4312, Crux Mathematicorum, Vol. 44, No. 2 (2018), pp. 69 and 71; Solution to Problem 4312, ibid., Vol. 45, No. 2 (2019), pp. 92-93.
Eric Weisstein's World of Mathematics, Bernoulli Polynomial.
Wikipedia, Bernoulli Polynomials.
Index entries for transcendental numbers.

Cf. A053382, A053383, A073747, A349003.

$MaxExtraPrecision = 1000; funs[n_] := BernoulliB[2 n, n]/n^(2 n); Do[Print[N[Sum[(-1)^(m + j)*funs[j*Floor[1000/m]] * j^(m - 1)/(j - 1)!/(m - j)!, {j, 1, m}], 110]], {m, 10, 100, 10}]
RealDigits[2/(E^2 - 1), 10, 110][[1]]

Equals 2/(exp(2)-1).
From Peter Luschny, Nov 05 2021: (Start)
Equals lim_{n->oo} (1/n) * Sum_{k=0..n-1} B(2*n, 1 + k/n) by J. L. Raabe's multiplication theorem.
Equals -2 * lim_{n->oo} HurwitzZeta(1 - 2*n, n) * n^(1 - 2*n). (End)
Equals A073747 - 1. - Alois P. Heinz, Nov 05 2021
Equals Sum_{k>=1} tanh(1/2^k)/2^k (Bell, 2018). - Amiram Eldar, Apr 12 2022

A280093 Pierce Expansion of coth(1).

Original entry on oeis.org

1, 3, 16, 38, 42, 139, 168, 385, 633, 942, 1728, 3017, 3842, 17453, 32989, 39408, 177334, 268130, 822437, 1522942, 3247926, 5937944, 22736433, 34285758, 51598089, 57736381, 105470828, 173010552, 541826347, 1758595979, 1803356572, 3331293851, 3545862229

Offset: 0

G. C. Greubel, Dec 25 2016

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Pierce Expansion
Wikipedia, Engel Expansion

Cf. A073747 (coth(1)).

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[Coth[1] , 7!], 50]

A307178 Decimal expansion of coth(1/2).

Original entry on oeis.org

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

Offset: 1

Terry D. Grant, Mar 27 2019

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

			2.163953413738... = 2 + 1/(6 + 1/(10 + 1/(14 + 1/(18 + ...)))).

Index entries for transcendental numbers

Cf. A016825 (continued fraction), A073333, A073747 (coth(1)).

Cf. A000367, A002445.

RealDigits[Coth[1/2], 10, 120][[1]] (* or *) BesselI[-1/2, 1/2]/BesselI[1/2, 1/2]

cotanh(1/2) \\ Michel Marcus, Mar 28 2019

Equals (exp(1)+1)/(exp(1)-1).
Equals (BesselI(3/2,1/2)/BesselI(1/2,1/2))+2.
Equals BesselI(-1/2,1/2)/BesselI(1/2,1/2).
Equals 2 * Sum_{k>=0} B(2*k)/(2*k)!, where B(2*k) = A000367(k)/A002445(k) are the Bernoulli numbers. - Amiram Eldar, Nov 25 2020
Equals 2 * A073333 + 1. - Antonio Graciá Llorente, Jan 21 2024

A346205 Decimal expansion of solution to LambertW(-x) - LambertW(-1,-x) = 2.

Original entry on oeis.org

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

Offset: 0

Gleb Koloskov, Jul 10 2021

			0.2288989481961786412366361253722...

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A072334, A073742, A073747.

SetDefaultRealField(RealField(135)); 2/(Exp(2)-1)*Exp(2/(1-Exp(2))); // G. C. Greubel, Jun 11 2024

x/.FindRoot[LambertW[-x]-LambertW[-1,-x]==2, {x, 0.1, 0.3}, WorkingPrecision -> 110]
RealDigits[2/(E^2-1)*Exp[2/(1-E^2)], 10, 135][[1]] (* G. C. Greubel, Jun 11 2024 *)

PARI
```
exp(-cotanh(1))/sinh(1)
```

numerical_approx(2/(e^2-1)*exp(2/(1-e^2)), digits=135) # G. C. Greubel, Jun 11 2024

Equals exp(-coth(1))/sinh(1) = exp(-A073747)/A073742.
Equals (coth(1)-1)*exp(1-coth(1)) = (A073747-1)*exp(1-A073747).
Equals (coth(1)+1)/exp(1+coth(1)) = (A073747+1)/exp(1+A073747).
Equals 2/(e^2-1)*exp(2/(1-e^2)) = 2/(A072334^2-1)*exp(2/(1-A072334^2)).

A384962 Decimal expansion of coth(2*Pi).

Original entry on oeis.org

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

Offset: 1

Jason Bard, Jun 13 2025

			1.0000069747090356162331216356036683646796758069002476...

Jason Bard, Table of n, a(n) for n = 1..10000
Michael I. Shamos, A catalog of the real numbers, (2007). See p. 20.

Cf. A019692, A073747, A175316.

Mathematica
```
RealDigits[Coth[2*Pi], 10, 100][[1]]
```

A279906 Decimal expansion of the number whose continued fraction expansion consists of the even numbers.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Dec 26 2016

			2.2401937238700897411052206417298297720272468672903936535447776204253890772...

Cf. A060997, A073747.

RealDigits[ FromContinuedFraction[ 2Range[ 38]], 10, 111]
RealDigits[ BesselI[0, 1]/BesselI[1, 1], 10, 111] (* Robert G. Wilson v, Feb 17 2017 *)

cp's OEIS Frontend

A109237 a(n) = floor(n*coth(1)).

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A298241 Decimal expansion of BesselI(1,2/3)/BesselI(0,2/3).

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A298243 Decimal expansion of BesselI(1,2/5)/BesselI(0,2/5).

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A036244 Denominator of continued fraction given by C(n) = [ 1; 3, 5, 7, ...(2n-1)].

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A349004 Decimal expansion of lim_{n->infinity} B(2*n, n)/n^(2*n), where B(n, x) is the n-th Bernoulli polynomial.

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A280093 Pierce Expansion of coth(1).

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Mathematica

A307178 Decimal expansion of coth(1/2).

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Formula

A346205 Decimal expansion of solution to LambertW(-x) - LambertW(-1,-x) = 2.

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A349004 Decimal expansion of lim_{n->infinity} B(2n, n)/n^(2n), where B(n, x) is the n-th Bernoulli polynomial.