A033262 -id:A033262 - 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

A033260 Continued fraction for Laplace's limit constant.

Original entry on oeis.org

0, 1, 1, 1, 27, 1, 1, 1, 8, 2, 154, 2, 4, 1, 5, 1, 1, 2, 1601, 2, 63, 1, 2, 12, 17, 9, 2, 1, 1, 57, 1, 6, 2, 4, 3, 10, 1, 6, 2, 1, 3, 4, 2, 1, 31, 3, 6, 1, 1, 4, 2, 1, 1, 1, 1, 1, 1, 14, 1, 4, 1, 2, 2, 1, 97, 1, 52, 2, 4, 1, 1, 5, 13, 1, 1, 3, 6, 2, 3, 3, 2, 2, 1, 1, 2, 28, 2, 3, 1, 4, 2, 1, 4

Offset: 0

Eric W. Weisstein

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 266-268.

G. C. Greubel, Table of n, a(n) for n = 0..20000
S. R. Finch, Laplace Limit Constant
Eric Weisstein's World of Mathematics, Laplace Limit

Cf. A033259 (decimal expansion), A033261, A033262, A033263.

ContinuedFraction[Sqrt[x^2-1]/.FindRoot[x==Coth[x], {x,1}, WorkingPrecision -> 250], 121] (* G. C. Greubel, Dec 13 2024 *)

Offset changed by Andrew Howroyd, Jul 04 2024

A033261 Position of first occurrence of n in the continued fraction for the Laplace's limit constant.

Original entry on oeis.org

1, 9, 34, 12, 14, 31, 100, 8, 25, 35, 101, 23, 72, 57, 750, 270, 24, 365, 363, 482, 191, 642, 821, 541, 393, 632, 4, 85, 2049, 617, 44, 201, 941, 182, 206, 862, 3104, 1295, 2122, 258, 1576, 5551, 158, 3353, 3870, 114, 506, 1669, 9646, 1127, 445, 66, 1804

Offset: 1

Eric W. Weisstein

nonn

The continued fraction expansion is indexed [a_0; a_1, a_2, a_3, ...].

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 266-268.

Steven R. Finch, Laplace Limit Constant [Broken link]
Steven R. Finch, Laplace Limit Constant [From the Wayback machine]
Eric Weisstein's World of Mathematics, Laplace Limit.

Cf. A033259, A033260, A033262, A033263.

A033260(a(n)) = n. - Andrew Howroyd, Sep 12 2024

More terms from Michel ten Voorde Jun 20 2003

Terms decreased by 1 for consistency with offset change in A033260 by Andrew Howroyd, Sep 12 2024

A033263 Positions of the incrementally largest terms in the continued fraction for Laplace's limit constant.

Original entry on oeis.org

1, 4, 10, 18, 1800, 2079, 6560, 7688, 11310, 12437, 24708, 63577

Offset: 1

Eric W. Weisstein

The continued fraction expansion is indexed [a_0; a_1, a_2, a_3, ...].

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 266-268.

Steven R. Finch, Laplace Limit Constant [Broken link]
Steven R. Finch, Laplace Limit Constant [From the Wayback machine]
Eric Weisstein's World of Mathematics, Laplace Limit.

Cf. A033259, A033260, A033261, A033262.

A033260(a(n)) = A033262(n). - Andrew Howroyd, Sep 12 2024

More terms from Michel ten Voorde Jun 20 2003

Terms decreased by 1 for consistency with offset change in A033260 by Andrew Howroyd, Sep 12 2024

cp's OEIS Frontend

A033259 Decimal expansion of Laplace's limit constant.

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A033260 Continued fraction for Laplace's limit constant.

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A033261 Position of first occurrence of n in the continued fraction for the Laplace's limit constant.

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Extensions

A033263 Positions of the incrementally largest terms in the continued fraction for Laplace's limit constant.

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Author

Keywords

Comments

References

Links

Crossrefs

Formula

Extensions