A085984 -id:A085984 - cp's OEIS frontend

A033259 Decimal expansion of Laplace's limit constant.

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

Offset: 0

Eric W. Weisstein

Maximum value taken by the function x/cosh(x), which occurs at A085984. - Hrothgar, Mar 12 2014
Given two equal coaxial circular rings of diameter D located in two parallel planes distant d apart, this constant is the maximum value of d / D so that there exists a catenoid resting on these two rings. - Robert FERREOL, Feb 07 2019
The maximum value of the eccentricity for which the Lagrange series expansion for the solution to Kepler's equation converges. Laplace (1827) calculated the value 0.66195. The Italian astronomer Francesco Carlini (1783 - 1862) found the limit 0.66 five years before Laplace (Sacchetti, 2020). - Amiram Eldar, Aug 17 2020

			0.662743419349181580974742097109252907056233549115022417520392534990971853086...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 266-268.
Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, p. 402.
John Oprea, The Mathematics of Soap Films: Explorations with Maple, Amer. Math. Soc., 2000, p. 183.

G. C. Greubel, Table of n, a(n) for n = 0..20000
Steven R. Finch, Laplace Limit Constant [Broken link]
Steven R. Finch, Laplace Limit Constant [From the Wayback machine]
J. J. Green, The Lipschitz constant for the radial projection on real l_p - implementation notes, 2012. - _N. J. A. Sloane_, Sep 19 2012
Pierre-Simon Laplace, Supplément au 5e volume du Traité de mécanique céleste, Paris (1827). See p. 11.
Simon Plouffe, The laplace limit constant(to 500 digits)
Andrea Sacchetti, Francesco Carlini: Kepler's equation and the asymptotic solution to singular differential equations, Historia Mathematica (2020), preprint, arXiv:2002.02679 [math.HO], 2020.
Eric Weisstein's World of Mathematics, Laplace Limit.
Eric Weisstein's World of Mathematics, Kepler's Equation.
Wikipedia, Laplace limit.
Index entries for transcendental numbers.

Cf. A033260, A033261, A033262, A033263, A085984.

x/.FindRoot[ x Exp[ Sqrt[ 1+x^2 ] ]/(1+Sqrt[ 1+x^2 ])==1, {x, 1} ]
Sqrt[x^2 - 1] /. FindRoot[ x == Coth[x], {x, 1}, WorkingPrecision -> 30 ] (* Leo C. Stein, Jul 30 2017 *)
RealDigits[Sqrt[Root[{# - (1 + #)/E^(2 #) - 1 &, 1.1996786}]^2 - 1], 10, 100][[1]] (* Eric W. Weisstein, Jul 15 2022 *)

sqrt(solve(u=1,2,tanh(u)-1/u)^2-1) \\ M. F. Hasler, Feb 01 2011

Equals sqrt(A085984^2-1). - Jean-François Alcover, May 14 2013

A209289 Number of functions f:{1,2,...,2n}->{1,2,...,2n} such that every preimage has an even cardinality.

Original entry on oeis.org

1, 2, 40, 2256, 250496, 46063360, 12665422848, 4866544707584, 2490379333697536, 1637285952230719488, 1344814260872574402560, 1349528279475362368847872, 1624638302165034485761966080, 2310920106523435237448955723776, 3834278385523271302103123693142016

Offset: 0

Geoffrey Critzer, Jan 16 2013

nonn

Note that the empty set has even cardinality.

			a(1) = 2 because there are 2 functions from {1,2} into {1,2} for which the preimage of both elements has even size: 1,1 (where the preimage of 1 is {1,2} and the preimage of 2 is the empty set) and 2,2 (where the preimage of 1 is the empty set and the preimage of 2 is {1,2}).

Seiichi Manyama, Table of n, a(n) for n = 0..210 (terms 0..80 from Alois P. Heinz)

Cf. A085984.

a:= n-> (2*n)! *coeff(series(cosh(x)^(2*n), x, 2*n+1), x, 2*n):
seq(a(n), n=0..20);  # Alois P. Heinz, Jan 19 2013

nn=32;Select[Table[n!Coefficient[Series[Cosh[x]^n,{x,0,nn}],x^n],{n,0,nn}],#>0&]
a[ n_] := If[ n < 0, 0, With[{m = 2 n}, m! SeriesCoefficient[ Cosh[x]^m, {x, 0, m}]]]; (* Michael Somos, Jul 02 2017 *)

{a(n) = if( n<0, 0, n=2*n; n! * polcoeff( cosh(x + x*O(x^n))^n, n))}; /* Michael Somos, Jul 02 2017 */

a(n) = (2n)! * [x^(2n)] cosh(x)^(2n).
a(n) = Sum_{i=0..2*n} (n-i)^(2*n)*binomial(2*n,i). - Vladimir Kruchinin, Feb 07 2013
a(n) ~ c * n^(2*n) * 2^(2*n) * (1-r)^(2*n) / ((2-r)^n * r^n * exp(2*n)), where r = 0.1664434403990353015638385297757806508596082... is the root of the equation (2/r-1)^(1-r) = exp(2), and c = 1.66711311920192939687232294044843869828... = 2/A085984. - Vaclav Kotesovec, Sep 03 2014, updated Mar 18 2024

A178255 Decimal expansion of (3+sqrt(17))/2.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, May 24 2010

Continued fraction expansion of (3+sqrt(17))/2 is A109007.
a(n) = A082486(n) for n > 1.
The rectangle R whose shape (i.e., length/width) is (3+sqrt(17))/2 can be partitioned into rectangles of shapes 3 and 3/2 in a manner that matches the periodic continued fraction [3, 3/2, 3, 3/2, ...].  R can also be partitioned into squares so as to match the periodic continued fraction [3, 1, 1, 3, 1, 1,...].  For details, see A188635. - Clark Kimberling, May 07 2011
The positive eigenvalue of the real symmetric 2 X 2 matrix M defined by M(i,j) = max(i,j) = [(1  2), (2  2)] is (3+sqrt(17))/2, while the negative one is (3-sqrt(17))/2. For a generalization, see A085984. - Bernard Schott, Apr 13 2020
A quadratic integer with minimal polynomial x^2 - 3x - 2. - Charles R Greathouse IV, Apr 14 2020
The positive root of x^2 - 3^x - 2. The negative root is -(-3 + sqrt(17))/2 = -0.56155...  - Wolfdieter Lang, Dec 10 2022

			(3+sqrt(17))/2 = 3.56155281280883027491...

Index entries for algebraic numbers, degree 2

Cf. A082486 (decimal expansion of (5+sqrt(17))/2), A010473 (decimal expansion of sqrt(17)), A109007 (repeat 3, 1, 1), A085984.

FromContinuedFraction[{3, 3/2, {3, 3/2}}]
ContinuedFraction[%, 100] (* [3,1,1,3,1,1,...] *)
RealDigits[N[%%, 120]]    (* A178255 *)
N[%%%, 40]
(* Clark Kimberling, May 07 2011 *)

(3+sqrt(17))/2 \\ Charles R Greathouse IV, Apr 14 2020

A009379 Expansion of log(1+tan(x)*x).

Original entry on oeis.org

0, 2, -4, 96, -2080, 125440, -8629248, 996007936, -140162633728, 27058965184512, -6350990843576320, 1866805063173799936, -653569786506324738048, 273136898848234632380416, -133034893921204302732328960

Offset: 0

R. H. Hardin

sign

Cf. A009399, A085984.

Log[ 1+Tan[ x ]*x ] (* Even Part *)
nn = 20; Table[(CoefficientList[Series[Log[1 + x*Tan[x]], {x, 0, 2*nn}], x] * Range[0, 2*nn]!)[[n]], {n, 1, 2*nn+1, 2}] (* Vaclav Kotesovec, Jan 23 2015 *)

a(n) ~ (2*n)! * (-1)^(n+1) / (n * r^(2*n)), where r = 1.1996786402577338339163698486411419442614587884... (see A085984) is the root of the equation r*tanh(r) = 1. - Vaclav Kotesovec, Jan 23 2015

Extended with signs by Olivier Gérard, Mar 15 1997

A009399 Expansion of log(1+tanh(x)*x).

Original entry on oeis.org

0, 2, -20, 576, -33312, 3258880, -485139456, 102300807168, -29028932390912, 10668077137133568, -4929291212351078400, 2797060130323340197888, -1912137417504544127975424, 1550018044651811766917922816

Offset: 0

R. H. Hardin

sign

Cf. A009379, A069855, A085984.

With[{nn=30},Take[CoefficientList[Series[Log[1+Tanh[x]*x],{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]] (* Harvey P. Dale, Jun 13 2016 *)

a(n) ~ -(-1)^n * sqrt(Pi) * 2^(2*n + 1) * n^(2*n - 1/2) / (r^(2*n) * exp(2*n)), where r = A069855 = 0.8603335890193797624838934241376623334118843632... is the root of the equation r * tan(r) = 1. - Vaclav Kotesovec, Dec 21 2017

Extended with signs by Olivier Gérard, Mar 15 1997

Prior Mathematica program replaced by Harvey P. Dale, Jun 13 2016

A069855 Decimal expansion of the root of x*tan(x)=1.

Original entry on oeis.org

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

Offset: 0

Benoit Cloitre, May 01 2002

Consider a lens-like shape S created by the curves cos(x) and -cos(x) for x in [-Pi/2,Pi/2] and the points A = (u, v), B = (-u, v), C = (-u, -v), D = (u, -v), K = (0, 2v), L = (-2u, 0), M = (0, -2v), N = (2u,0), where u is given by this sequence, and v = u/sqrt(1+u^2). Then ABCD is the rectangle of maximal area, inscribed in S, with sides parallel to the coordinate axes, and KLMN is the rhombus of minimal area, circumscribed around S, with vertices on the coordinate axes. Also, A,B,C,D are the tangent points where the sides of the rhombus touch S, see illustration in the links section. - Gleb Koloskov, Jul 05 2021

			0.860333589019379762483893424137662333411884363237653783...

Gleb Koloskov, Geometric illustration
Eric Weisstein's World of Mathematics, Cotangent [From _Eric W. Weisstein_, Mar 03 2010]

Cf. A009399, A085984, A346062.

N[Minimize[{(x+Cot[x])^2 Sin[x],{x>0,xGleb Koloskov, Jul 05 2021 *)
RealDigits[x/.FindRoot[x Tan[x]==1,{x,1},WorkingPrecision->120]][[1]] (* Harvey P. Dale, Dec 04 2021 *)

/* 300 significant digits */ s=0.1; for(n=1,500,s=s+sign(cotan(s)-s)/2^n; if(n>499, print(s*1.)))

Equals A346062 * sqrt(2 + 2*sqrt(1 + 256/A346062^2)) / 16. - Gleb Koloskov, Jul 05 2021

A248916 Decimal expansion of gamma = 8*lambda^2, a critical threshold of a boundary value problem, where lambda is Laplace's limit constant A033259.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Oct 16 2014

The boundary value problem y''(x) + c*exp(y(x)) = 0, y(0) = y(1) = 0 and c > 0, has 0, 1 or 2 solutions when c > gamma, c = gamma and c < gamma, respectively. [After Steven Finch]

			3.5138307191251612062078370932388235871...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 4.8 Laplace limit constant, p. 266.

G. C. Greubel, Table of n, a(n) for n = 1..20000
Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 32.
Eric Weisstein's MathWorld, Laplace Limit

Cf. A033259, A085984.

digits = 104; lambda = x /. FindRoot[x Exp[Sqrt[1 + x^2]]/(1 + Sqrt[1 + x^2]) == 1, {x, 1}, WorkingPrecision -> digits + 5]; gamma = 8*lambda^2; RealDigits[gamma, 10, digits] // First

A249136 Decimal expansion of the largest constant 'beta' for which there exists a solution to the differential equation y''(x)+exp(y(x))=0, with y(0)=y(beta)=0.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Oct 22 2014

			1.874521464034264187600324820470264120147219398917056...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020-2024, p. 32.
Eric Weisstein's MathWorld, Laplace Limit.

Cf. A033259, A085984, A248916.

lambda = x /. FindRoot[x*Exp[Sqrt[1 + x^2]]/(1 + Sqrt[1 + x^2]) == 1, {x, 1}, WorkingPrecision -> 102]; beta = Sqrt[8]*lambda; RealDigits[beta] // First

beta = sqrt(8)*lambda, where lambda is A033259, the Laplace limit constant 0.66274...

Equals sqrt(A248916). - Hugo Pfoertner, Dec 19 2024

A345737 Decimal expansion of the initial angle in radians above the horizon that maximizes the length of a projectile's trajectory.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Jun 25 2021

A projectile is launched with an initial speed v at angle theta above the horizon. Assuming that the gravitational acceleration g is uniform and neglecting the air resistance, the trajectory is a part of a parabola whose length is maximized when the angle is the root of the equation csc(theta) = coth(csc(theta)). The maximal length is then u * v^2/g, where u = 1.1996... is the root of coth(x) = x (A085984).
The angle in degrees is 56.4658351274...
The initial angle that maximizes the horizontal distance is the well-known result theta = Pi/4 = 45 degrees. The corresponding length of trajectory in this case is u * v^2/g, where u = (sqrt(2) + arcsinh(1))/2 = 1.1477... (A103711), which is 95.67...% of the maximum value.

			0.98551473786231546211492853725730463877247220596742...

Thomas Szirtes, Applied Dimensional Analysis and Modeling, Butterworth-Heinemann, 2007, p. 578.

Joshua Cooper and Anton Swifton, Throwing a ball as far as possible, revisited, The American Mathematical Monthly, Vol. 124, No. 10 (2017), pp. 955-959; arXiv preprint, arXiv:1611.02376 [math.HO], 2016.
Haiduke Sarafian, On projectile motion, The Physics Teacher, Vol. 37, No. 2 (1999), pp. 86-88.
Ju Yan-Qing, Projectile motion path length and initial projectile angle, Journal of Science of Teachers' College and University, Vol. 3 (2005), pp. 49-51.

Cf. A033259, A085984, A103711, A240358, A345738, A345739.

Digits:=100:fsolve(tan(x)=sinh(csc(x)),x=0..1); (# Robert FERREOL, Jun 17 2025)

RealDigits[ArcCsc[x /. FindRoot[x == Coth[x], {x, 1}, WorkingPrecision -> 120]], 10, 100][[1]]

solve(x=0,1,my(s=sin(x)); s*atanh(s)-1) \\ Charles R Greathouse IV, Sep 18 2024

asin(solve(u=.5, 1, tanh(1/u)-u)) \\ Charles R Greathouse IV, Sep 18 2024

Equals arccsc(u) where u is the root of coth(x) = x (A085984).
Equals arctan(A240358) = arctan(1/A033259). - Robert FERREOL, Jun 16 2025
Positive root of tan(x) = sinh(csc(x)). - Robert FERREOL, Jun 17 2025

A140133 Decimal expansion of the area enclosed in the lens-shaped region of the Laplace Limit.

Original entry on oeis.org

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

Offset: 1

Jonathan Vos Post, Jun 04 2008

See Weisstein for complex analysis function.

			1.8532684487079870332219364034397278879469653896325464...

G. C. Greubel, Table of n, a(n) for n = 1..3000
Eric W. Weisstein, Laplace Limit (value given is incorrect)

Cf. A033259, A085984.

f[x_] := (Sqrt[x - Tanh[x]]*(x*Csch[x]^2 + 2*x - Coth[x]))/(2* Sqrt[-x + Coth[x]]); xmax = x /. FindRoot[Coth[x] - x == 0, {x, 1}, WorkingPrecision -> 200]; First[ RealDigits[ Chop[ Quiet[ NIntegrate[f[x], {x, 0, xmax}, WorkingPrecision -> 200, MaxRecursion -> 20]]*4], 10, 100]] (* Jean-François Alcover, Jun 07 2012, after D. S. McNeil *)

def A140133_cons(dps=200):
    from mpmath import mp, sqrt, tanh, coth, csch, findroot, quad
    mp.dps = 2*dps # safety
    def f(x): return 1/2*sqrt(x - tanh(x))*(x*csch(x)^2 + 2*x - coth(x))/sqrt(-x + coth(x))
    xmax = findroot(lambda x: coth(x)-x, 1)
    return quad(f, [0, xmax])*4  # D. S. McNeil, Feb 01 2011

cp's OEIS Frontend

A033259 Decimal expansion of Laplace's limit constant.

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A209289 Number of functions f:{1,2,...,2n}->{1,2,...,2n} such that every preimage has an even cardinality.

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A178255 Decimal expansion of (3+sqrt(17))/2.

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A009379 Expansion of log(1+tan(x)*x).

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Mathematica

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A009399 Expansion of log(1+tanh(x)*x).

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Mathematica

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A069855 Decimal expansion of the root of x*tan(x)=1.

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A248916 Decimal expansion of gamma = 8*lambda^2, a critical threshold of a boundary value problem, where lambda is Laplace's limit constant A033259.

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Mathematica

A249136 Decimal expansion of the largest constant 'beta' for which there exists a solution to the differential equation y''(x)+exp(y(x))=0, with y(0)=y(beta)=0.