A173272 Decimal expansion of the positive solution of sqrt((2-x)(2+x)) + sqrt((3-x)(3+x)) = sqrt((2-x)(2+x))*sqrt((3-x)(3+x)).

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

Offset

1,2

Comments

x is the solution to the problem of the "crossed ladders" where the height at which they cross is 1, the ladder lengths 2 and 3, and the distance between the walls 'x'.

x = sqrt(4-y^2) = 1.23118572...;

y = (1 + sqrt(z-4) + sqrt(2*sqrt(4+z^2) - 2*sqrt(z-4) - z - 3))/2 = 1.57612871...

with z = (5/3) + ((395/27) + sqrt(5200/27))^(1/3) + ((395/27) - sqrt(5200/27))^(1/3) = 5.63079775...

A root of the polynomial x^8 - 22*x^6 + 163*x^4 - 454*x^2 + 385. [R. J. Mathar, Feb 21 2010]

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

Wikipedia, Problème des deux échelles (in French). [Robert FERREOL, Jul 26 2022]

Example

1.231185723778668829962705... [R. J. Mathar, Feb 21 2010]

Maple

Digits := 120 ; fsolve(x^8-22*x^6+163*x^4-454*x^2+385, x, 1.1..1.3) ; # R. J. Mathar, Feb 21 2010

Mathematica

Root[#^8 - 22#^6 + 163#^4 - 454#^2 + 385 &, 3] // RealDigits[#, 10, 105]& // First (* Jean-François Alcover, Feb 22 2013 *)
RealDigits[x/.FullSimplify[With[{a=Sqrt[(2-x)(2+x)], b=Sqrt[(3-x)(3+x)]}, Solve[a*b==a+b, x]]][[2]], 10, 120][[1]] (* Essentially identical to Jean- Francois Alcover's program above *) (* Harvey P. Dale, Dec 26 2014 *)

Crossrefs

Sequence in context: A217897 A135900 A338072 * A326303 A047789 A068869

Adjacent sequences: A173269 A173270 A173271 * A173273 A173274 A173275

Keyword

cons,nonn

Author

Philippe Deléham, Feb 14 2010

Extensions

More digits from R. J. Mathar, Feb 21 2010

Status

approved