A237558 Decimal expansion of (1/Pi)*arccos(1/sqrt(3)).

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

Offset

0,1

Comments

Known to be irrational. If m is a positive integer (1/Pi)*arccos(1/sqrt(m)) is rational iff m=1,2 or 4.

A constant giving the location (x = y) of the thermodynamic center (also called warmest point or "hot spot") of a unit isosceles right triangle. The thermodynamic center is the warmest point in a convex heat conductor with unitary initial temperature and with boundary grounded at zero temperature. The convex heat conductor here considered is the isosceles right triangle x>0, y>0, x+y <= 1. - Jean-François Alcover, Jul 26 2016

References

M. Aigner and G. M. Ziegler, Proofs from The Book, Chap. 7, p. 40, Springer-Verlag, Berlin, 1999.

Links

Table of n, a(n) for n=0..98.

Steven R. Finch, In limbo: Three triangle centers, arXiv:1406.0836 [math.HO] 2014 p. 11.

Formula

0 < x < 1/2 maximizing sin(Pi x)*sin(2Pi x), that is x = arctan(sqrt(2)) / Pi, solution to sin(Pi x) = 3 sin(3 Pi x). - Jean-François Alcover, Jul 26 2016

Example

0.30408672...

Maple

arccos(1/sqrt(3))/Pi; evalf(%) ; # R. J. Mathar, Aug 02 2016

Mathematica

Join[{0}, RealDigits[1/Pi ArcCos[1/Sqrt[3]], 10, 120][[1]]] (* Harvey P. Dale, Feb 22 2015 *)

Crossrefs

Cf. A275336.

Sequence in context: A371737 A173425 A289445 * A060034 A308216 A035544

Adjacent sequences: A237555 A237556 A237557 * A237559 A237560 A237561

Keyword

nonn,cons

Author

Benoit Cloitre, Feb 09 2014

Status

approved