A085984 -id:A085984

A033259

Decimal expansion of Laplace's limit constant.

+10
12

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

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

REFERENCES

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.

LINKS

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.

FORMULA

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

EXAMPLE

0.662743419349181580974742097109252907056233549115022417520392534990971853086...

MATHEMATICA

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 *)

PROG

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

CROSSREFS

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

KEYWORD

nonn,cons,changed

AUTHOR

Eric W. Weisstein

STATUS

approved

A178255

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

+10
6

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

Table of n, a(n) for n=1..105.

Index entries for algebraic numbers, degree 2

EXAMPLE

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

MATHEMATICA

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 *)

PROG

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

CROSSREFS

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

KEYWORD

cons,nonn

AUTHOR

Klaus Brockhaus, May 24 2010

STATUS

approved

A009379

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

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..14.

FORMULA

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

MATHEMATICA

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 *)

CROSSREFS

Cf. A009399, A085984.

KEYWORD

sign

AUTHOR

R. H. Hardin

EXTENSIONS

Extended with signs by Olivier Gérard, Mar 15 1997

STATUS

approved

A009399

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

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..13.

FORMULA

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

MATHEMATICA

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 *)

CROSSREFS

Cf. A009379, A069855, A085984.

KEYWORD

sign

AUTHOR

R. H. Hardin

EXTENSIONS

Extended with signs by Olivier Gérard, Mar 15 1997

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

STATUS

approved

A069855

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

+10
4

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

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

LINKS

Table of n, a(n) for n=0..103.

Gleb Koloskov, Geometric illustration

Eric Weisstein's World of Mathematics, Cotangent [From Eric W. Weisstein, Mar 03 2010]

FORMULA

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

EXAMPLE

0.860333589019379762483893424137662333411884363237653783...

MATHEMATICA

N[Minimize[{(x+Cot[x])^2 Sin[x], {x>0, x<Pi/2}}, x][[2]], 300][[1]][[2]] (* Gleb Koloskov, Jul 05 2021 *)

RealDigits[x/.FindRoot[x Tan[x]==1, {x, 1}, WorkingPrecision->120]][[1]] (* Harvey P. Dale, Dec 04 2021 *)

PROG

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

CROSSREFS

Cf. A009399, A085984, A346062.

KEYWORD

cons,easy,nonn

AUTHOR

Benoit Cloitre, May 01 2002

STATUS

approved

A209289

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

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Note that the empty set has even cardinality.

LINKS

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

FORMULA

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

EXAMPLE

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}).

MAPLE

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

MATHEMATICA

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 *)

PROG

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

CROSSREFS

Cf. A085984.

KEYWORD

nonn

AUTHOR

Geoffrey Critzer, Jan 16 2013

STATUS

approved

A248916

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

+10
3

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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]

REFERENCES

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

LINKS

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

EXAMPLE

3.5138307191251612062078370932388235871...

MATHEMATICA

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

CROSSREFS

Cf. A033259, A085984.

KEYWORD

nonn,cons,easy,changed

AUTHOR

Jean-François Alcover, Oct 16 2014

STATUS

approved

A345737

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

+10
2

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

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.

REFERENCES

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

LINKS

Table of n, a(n) for n=0..86.

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.

FORMULA

Equals arccsc(u) where u is the root of coth(x) = x (A085984).

EXAMPLE

0.98551473786231546211492853725730463877247220596742...

MATHEMATICA

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

PROG

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

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

CROSSREFS

Cf. A085984, A103711, A345738, A345739.

KEYWORD

nonn,cons

AUTHOR

Amiram Eldar, Jun 25 2021

STATUS

approved

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.

+10
1

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..102.

Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 32.

Eric Weisstein's MathWorld, Laplace Limit.

FORMULA

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

EXAMPLE

1.874521464034264187600324820470264120147219398917056...

MATHEMATICA

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

CROSSREFS

Cf. A033259, A085984, A248916.

KEYWORD

nonn,cons,easy

AUTHOR

Jean-François Alcover, Oct 22 2014

STATUS

approved

A345736

Decimal expansion of the surface area of the catenoid-shaped soap film between two parallel coaxial unit-radius circular rings whose distance between their centers is maximal (2*A033259).

+10
1

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..20000

Markus Deserno, Notes on Differential Geometry, with special emphasis on surfaces in R^3, 2004. See p. 28.

Lev A. Slobozhanin, J. Iwan, D. Alexander and Viral D. Patel, The stability margin for stable weightless liquid bridges, Physics of Fluids, Vol. 14, No. 1 (2002), pp. 209-224. See p. 222.

Eric Weisstein's World of Mathematics, Minimal Surface of Revolution.

FORMULA

Equals 2*Pi*A085984.

EXAMPLE

7.53780320580457797882651822069283034551193592152611...

MATHEMATICA

RealDigits[2 * Pi * x /. FindRoot[x == Coth[x], {x, 1}, WorkingPrecision -> 120], 10, 100][[1]]

CROSSREFS

Cf. A033259, A085984, A253545.

KEYWORD

nonn,cons,changed

AUTHOR

Amiram Eldar, Jun 25 2021

STATUS

approved