A354684 Decimal expansion of the horizontal distance between the equal-height endpoints of a suspended unit-length chain for which the area between the chord joining the endpoints and the chain has a maximum value.

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

Offset

0,1

Comments

The maximum area is x*/(2*(x^2 + 2)) = 0.154908..., where x is given in the Formula section.

Links

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

Amiram Eldar, Illustration.

Murray S. Klamkin, Problem E 1199, The American Mathematical Monthly, Vol. 63, No. 1 (1956), p. 39; Maximum Area Between a Hanging Chain and Its Chord, Solution to Problem E 1199 by C. M. Sandwick, Sr., ibid., Vol. 63, No. 7 (1956), pp. 495-496.

Wikipedia, Catenary.

Formula

Equals arcsinh(x)/x where x = 2.391374... is the positive root of the equation arcsinh(x)/x = 2*sqrt(x^2+1)/(x^2+2).

Example

0.67162853059005963017701121889676242415980602507067...

Mathematica

xmax = x /. FindRoot[ArcSinh[x]/x == 2*Sqrt[x^2 + 1]/(x^2 + 2), {x, 2}, WorkingPrecision -> 120]; RealDigits[ArcSinh[xmax]/xmax][[1]]

Crossrefs

Cf. A225146.

Sequence in context: A011483 A319458 A296459 * A308915 A294644 A256128

Adjacent sequences: A354681 A354682 A354683 * A354685 A354686 A354687

Keyword

nonn,cons

Author

Amiram Eldar, Jun 03 2022

Status

approved