cp's OEIS Frontend

Average length of chord formed from two randomly chosen points on the circumference of a unit circle (see Weisstein/MathWorld link). - Rick L. Shepherd, Jun 19 2006

Suppose u(0) = 1 + i where i^2 = -1 and u(n+1) = (1/2)*(u(n) + |u(n)|). Conjecture: lim_{n -> oo} Real(u(n)) = 4/Pi. - Yalcin Aktar, Jul 18 2007

Ratio of the arc length of the cycloid for one period to the circumference of the corresponding circle rolling on a line. Thus, for any integral number n of revolutions of a circle of radius r, a point on the circle travels (4/Pi)*2*Pi*r*n = 8*r*n (while the center of the circle moves only 2*Pi*r*n). This ratio varies for partial revolutions and depends upon the initial position of the point with points nearest the line moving the slowest (see Dudeney, who explains how the tops of bicycle wheels move faster than the parts nearest the ground). - Rick L. Shepherd, May 05 2014

Average distance traveled in two steps of length 1 for a random walk in the plane starting at the origin. - Jean-François Alcover, Aug 04 2014

Ratio of the circle area to the area of a square having equal perimeters. - Iaroslav V. Blagouchine, May 06 2016

This is also the value of a special case (n=1) of an n-family of series considered by Hardy (see A278145): 1 + (1/2)*(1/2)^2 + (1/3)*(1*3/(2*4))^2 + (1/4)*((1*3*5) / (2*4*6))^2 + ... = Sum_{k>=0} (1/(k+1))*((2*k-1)!!/(2*k)!!)^2. - Wolfdieter Lang, Nov 14 2016

Minimum ratio of the area of a rectangle to one of its inscribed ellipses, or only existing ratio if the rectangle is a square and then the only inscribed ellipse is a circle. This ellipse has its semiaxes parallel to the sides of the rectangle. If a rectangle has sides of length 2a and 2b, its area is 4*a*b, while the ellipse inscribed has a and b as semiaxes, therefore its area is a*b*Pi. Thus the ratio is (4*a*b)/(a*b*Pi) = 4/Pi. - Giovanni Zedda, Jun 20 2019

The diameter of the conventional spherical earth is (4/Pi)*10000 km = 12732.395... km. - Jean-François Alcover, Oct 30 2021

From Jianing Song, Aug 06 2022: (Start)

sign(sin(x)) = (4/Pi) * Sum_{n>=0} sin((2*n+1)*x)/(2*n+1), for all x in R;

sign(cos(x)) = (4/Pi) * Sum_{n>=0} (-1)^n*cos((2*n+1)*x)/(2*n+1), for all x in R. (End)

6/Pi = Volume of the cuboid (If L1>L2>L3) / Volume of the inscribed ellipsoid.

6/Pi = Volume of the cuboid (If L1>(L2=L3)) / Volume of the inscribed spheroid.

6/Pi = Volume of the regular hexahedron (or cube) / Volume of the inscribed Sphere.

6/Pi = 1 / Arc of 30 degrees.

6/Pi = Volume of the cuboid (If L1<(L2=L3)) / Volume of the inscribed spheroid.

6/Pi = Surface area of the regular hexahedron (or cube) / surface area of the inscribed sphere.

The preferred order for these five numbers is 4, 6, 8, 12, 20 (tetrahedron, octahedron, cube, icosahedron, dodecahedron), as in A053016. - N. J. A. Sloane, Nov 05 2020

Also number of faces of Platonic solids ordered by increasing ratios of volumes to their respective circumscribed spheres. See cross-references for actual ratios. - Rick L. Shepherd, Oct 04 2009

Also the expected lengths of nontrivial random walks along the edges of a Platonic solid from one vertex back to itself. - Jens Voß, Jan 02 2014

From Bernard Schott, Apr 17 2022: (Start)

For any triangle ABC, (see Crux Mathematicorum):

(b+c)/A + (c+a)/B + (a+b)/C >= (12/Pi) * s,

b*c/(A*(s-a)) + c*a/(B*(s-b)) + a*b/(C*(s-c)) >= (12/Pi) * s,

where (A,B,C) are the angles (measured in radians), (a,b,c) the side lengths of this triangle and s the semiperimeter.

Equality stands iff triangle ABC is equilateral. (End)

9/Pi = 2.864788975654...

The equivalent sequence for Zeta(3) repeated after very few terms. When, if ever, does this sequence start to repeat?

Original definition: Limit of (1/Pi)^...^(1/Pi), n times, as n approaches infinity. Equals exp(-LambertW(log(Pi))).

The value can be obtained by iterating x -> 1/Pi^x with any real starting value, but convergence is linear and slow: about 5 iterations are needed for each additional decimal digit. - M. F. Hasler, Nov 01 2011

According to the Weisstein link, infinite iterated exponentiation such as used here, which is referred to both as an "infinite power tower" and "h(x)" -- with graph and other notations -- "converges iff e^(-e) <= x <= e^(1/e) as shown by Euler (1783) and Eisenstein (1844)" (citing Le Lionnais and Wells references). e^(-e) = A073230. e^(1/e) = A073229. x of interest here = 1/Pi = A049541. (1/A073243)^(1/A073243) = A030437^A030437 = Pi.

If y = h(x) = x^x^x^... converges, then by substitution y = x^y. So x^x^x^... is a solution y to the equation y^(1/y) = x. - Jonathan Sondow, Aug 27 2011

The expressions involving "..." in the above comment are misleading, since the limit is not obtained by applying additional "^x" to the previous expression, i.e., iterating "t -> t^x", but corresponds to iterations of "t -> x^t". - M. F. Hasler, Nov 01 2011

The asymptotic density of squarefree numbers that are divisible by 5. - Amiram Eldar, Mar 25 2021