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
KEYWORD
cons,nonn
AUTHOR
Philippe Deléham, Feb 14 2010
EXTENSIONS
More digits from R. J. Mathar, Feb 21 2010
STATUS
approved