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A253545 Decimal expansion of r = 0.527697..., a boundary ratio separating catenoid and Goldschmidt solutions in the minimal surface of revolution problem. 0
5, 2, 7, 6, 9, 7, 3, 9, 6, 9, 6, 2, 5, 7, 1, 5, 2, 8, 5, 7, 2, 4, 2, 3, 3, 4, 3, 3, 6, 3, 1, 8, 0, 5, 7, 7, 9, 6, 8, 8, 5, 3, 7, 9, 0, 6, 3, 1, 4, 1, 9, 5, 4, 1, 7, 2, 2, 2, 7, 5, 1, 5, 9, 5, 0, 1, 6, 2, 0, 7, 6, 8, 3, 2, 4, 5, 1, 9, 8, 8, 4, 4, 6, 6, 8, 4, 5, 2, 9, 3, 6, 0, 0, 5, 4, 7, 5, 3, 0, 3, 5, 1, 4, 1, 5 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Consider two circular frames each of diameter D and with a separation of d.

If d/D < r = 0.527697..., then a catenoid gives the absolute minimum area.

If r < d/D < L = 0.66274... (Laplace limit), there are 3 minimal surfaces of revolution passing through the frames: 2 catenoids and the so-called Goldschmidt discontinuous solution consisting of the 2 disks.

If d/D > L, there remains only the Goldschmidt solution.

LINKS

Table of n, a(n) for n=0..104.

Robert Ferréol's MathCurve, Catenoid

Eric Weisstein's MathWorld, Laplace Limit

Eric Weisstein's MathWorld, Minimal Surface of Revolution

FORMULA

arccosh(u)/u, where u = 1.21136... is solution to u*sqrt(u^2-1) + arccosh(u) - u^2 = 0.

Solution of 2*cosh((x^2+1)/2) = x+1/x. - Robert FERREOL, Feb 07 2019

EXAMPLE

0.5276973969625715285724233433631805779688537906314195417222751595...

MATHEMATICA

digits = 105; u0 = u /. FindRoot[u*Sqrt[u^2-1] + ArcCosh[u] - u^2 == 0, {u, 6/5}, WorkingPrecision -> digits+5];  r = ArcCosh[u0]/u0; RealDigits[r, 10, digits] // First

CROSSREFS

Cf. A033259 (Laplace limit).

Sequence in context: A250720 A080350 A204899 * A195343 A074454 A256110

Adjacent sequences:  A253542 A253543 A253544 * A253546 A253547 A253548

KEYWORD

nonn,cons,easy

AUTHOR

Jean-François Alcover, Apr 21 2015

STATUS

approved

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Last modified July 6 02:21 EDT 2020. Contains 335475 sequences. (Running on oeis4.)