A391410 - OEIS

%I #14 Apr 19 2026 05:40:00

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

%T 7,1,1,2,3,5,7,8,1,5,2,3,3,8,1,7,5,8,3,5,4,5,6,9,9,2,7,6,1,5,4,1,5,8,

%U 4,5,2,3,8,9,2,5,8,0,5,5,1,0,0,1,6,6,8,4,9,3,4

%N Decimal expansion of the smallest positive root of P(x) = 2*x^4 + 36*x^3 - 22*x^2 - 6*x + 2.

%C This is the scaled distance x=a/L of the 2 support points of a beam of length L from the endpoints of the beam. For a horizontal beam subject to gravity, this support maximizes the distance measured between the two endpoints of the beam's neutral fiber.

%C Friedrich Wilhelm Bessel wrote "Der Stab verkürzt sich also am wenigsten, wenn man ihn 0.22031 seiner ganzen Länge, von den Endpunkten entfernt, auflegt." (see page 132 of the reference).

%D Friedrich Wilhelm Bessel, Darstellung der Untersuchungen und Maaßregeln, welche, in 1835 bis 1838, durch die Einheit des Preußischen Längenmaaßes veranlaßt worden sind. Beilage I. Einfluss der Schwere auf die Figur eines, auf zwei Punkten von gleicher Höhe aufliegenden Stabes, Berlin, 1839, page 132.

%H Friedrich Geyer, <a href="https://web.archive.org/web/20210128041736/http://www.excelution.at/bessel-punkte/">Bessel-Punkte</a>, 13 Jan 2015.

%H Wikimedia, <a href="https://upload.wikimedia.org/wikipedia/commons/6/6a/Beam_supported_at_Bessel_points.png">Beam supported at Bessel points</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Airy_points#Bessel_points">Airy points</a>.

%H <a href="/index/Al#algebraic_04">Index entries for algebraic numbers, degree 4</a>.

%e 0.220309940683478575359621124864051...

%t RealDigits[Root[2*x^4 + 36*x^3 - 22*x^2 - 6*x + 2, 3], 10, 120][[1]] (* _Amiram Eldar_, Apr 19 2026 *)

%o (PARI) polrootsreal(2*x^4+36*x^3-22*x^2-6*x+2)[3]

%o (PARI) solve(x=0.2, 0.3, 2*x^4+36*x^3-22*x^2-6*x+2)

%Y Cf. A389416.

%K nonn,cons

%O 0,1

%A _Hugo Pfoertner_, Dec 09 2025