A225146 Decimal expansion of difference in area between a parabola and a catenary up to the cross-over point (at x=1 with vertex at y=0).

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

Offset

0,4

Comments

This difference far exceeds the difference between the parabola and the quarter arc of a circle.

Catenary - Shape of a cable in its natural state between two points.

Funicular - The shape of a cable under a load or loads between two points.

Parabola - The shape of a cable between two points under uniform load.

Arc - The shape of a cable between two points being at a constant distant from a third point.

The area under the Parabola minus the area under the Catenary.

Links

Table of n, a(n) for n=0..131.

David Griffin, Catenaries, Parabolas and Suspension Bridges

Math Forum, Ask Dr. Math, Catenary and Parabola Comparison

Wikipedia, Catenary

Wikipedia, Parabola

Formula

Equals 2/(e-1)^2 - 2/3. - Jean-François Alcover, Feb 18 2014

Example

0.010727108010265122454156368548309422002905984483643365507680033...

Maple

evalf(2/(exp(1)-1)^2-2/3, 150);  # Alois P. Heinz, Jul 16 2021

Mathematica

(* focus at y=0 *) NIntegrate[x^2 - (Cosh[x] - 1)/(Cosh[1] - 1), {x, 0, 1}, WorkingPrecision -> 2^7, MinRecursion -> 2^10, MaxRecursion -> 2^12]
(* to view the three curves, parabola, catenary and a ¼ arc of a circle *) Plot[{(Cosh[x] - 1)/(Cosh[1] - 1), x^2, -Sqrt[-x^2 + 1] + 1}, {x, 0, 1}]
Join[{0}, RealDigits[2/(E-1)^2-2/3, 10, 128][[1]]] (* Jean-François Alcover, Feb 18 2014 *)

Crossrefs

Sequence in context: A248284 A350620 A195404 * A021583 A328904 A352301

Adjacent sequences: A225143 A225144 A225145 * A225147 A225148 A225149

Keyword

nonn,cons

Author

Robert G. Wilson v, Aug 14 2013

Extensions

a(128) corrected by Georg Fischer, Jul 16 2021

Status

approved