

A225146


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


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, 8
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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..128.
David Griffin, Catenaries, Parabolas and Suspension Bridges
Math Forum, Ask Dr. Math, Catenary and Parabola Comparison
Wikipedia, Catenary
Wikipedia, Parabola


FORMULA

Equals 2/(e1)^2  2/3. [JeanFrançois Alcover, Feb 18 2014]


EXAMPLE

= 0.010727108010265122454156368548309422002905984483643365507680033…


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/(E1)^22/3, 10, 128][[1]]] (* JeanFrançois Alcover, Feb 18 2014 *)


CROSSREFS

Sequence in context: A208852 A248284 A195404 * A021583 A328904 A177437
Adjacent sequences: A225143 A225144 A225145 * A225147 A225148 A225149


KEYWORD

nonn,cons


AUTHOR

Robert G. Wilson v, Aug 14 2013


STATUS

approved



