A358981 -id:A358981 - cp's OEIS frontend

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

Offset: 0

Joshua Searle, Oct 30 2024

Given two unit 4 dimensional hyperspheres whose centers are a unit distance apart, this value is the fraction of their surface that lies within the other. 4 dimensions is the smallest number for this not to be a rational number.
The general formula in n dimensions is given by (1/2) * I(3/4; (n-1)/2, 1/2) where I(x; a, b) is the regularized incomplete beta function.
Let f(n) = 1/2 * I(3/4; (n-1)/2, 1/2).
  f(1) = 1/2
  f(2) = 1/3
  f(3) = 1/4
  f(4) = 1/3 - sqrt(3)/(4*Pi)
  f(5) = 5/32
  f(6) = 1/3 - 3*sqrt(3)/(8*Pi)
Odd n give a fraction with a denominator being a power of 2 and even n give 1/3 - k*sqrt(3)/Pi where k is a rational number (see SeqFans Mailing List Thread).
As n -> infinity, f(n) -> 0. This lack of overlap is a demonstration of why higher dimensional spheres are referred to as being 'spiky', despite being entirely convex.
The fact that f(n) -> 0 as n -> oo implies that the sequence of rationals k converges to Pi/sqrt(3^3) = A073010. - M. F. Hasler, Oct 31 2024

			0.1955011094778853209555017087550909729839867132416731701334918282605757574660158846...

Joshua Searle and others, Overlapping Hyperspheres, discussion in SeqFan mailing list, Oct 29, 2024.
Wikipedia, Spherical Cap
Index entries for transcendental numbers

Cf. A377523 (continued fraction), A102519, A132116, A258147, A343235, A358981.

RealDigits[1/3 - Sqrt[3]/(4*Pi), 10, 120][[1]] (* Amiram Eldar, Oct 31 2024 *)

A377522_upto(n)={localprec(n+++9);digits(10^n*(4/3-sqrt(3)/Pi)\4)} \\ M. F. Hasler, Oct 31 2024

from sympy import S, betainc_regularized, hyper
f = lambda n: betainc_regularized(S(n-1)/2, S(1)/2, 0, S(3)/4).rewrite(hyper).gammasimp().simplify().expand()/2
f(4).n(99) # decimal expansion
[int(d) for d in str(_)[2:-1]] # digits, omitting "0." and last (could be rounded)
# M. F. Hasler, Oct 31 2024

Equals (1/2) * I(3/4; 3/2, 1/2).

Equals A102519/3 = A258147/2 = A358981/Pi. - Hugo Pfoertner, Oct 31 2024

cp's OEIS Frontend

A377522 Decimal expansion of 1/3 - sqrt(3)/(4*Pi).

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