A377522 - OEIS

%I #36 Nov 21 2024 09:05:10

%S 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,

%T 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,

%U 5,7,5,7,4,6,6,0,1,5,8,8,4,6,2,3,2,3,8

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

%C 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.

%C 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.

%C Let f(n) = 1/2 * I(3/4; (n-1)/2, 1/2).

%C   f(1) = 1/2

%C   f(2) = 1/3

%C   f(3) = 1/4

%C   f(4) = 1/3 - sqrt(3)/(4*Pi)

%C   f(5) = 5/32

%C   f(6) = 1/3 - 3*sqrt(3)/(8*Pi)

%C 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).

%C 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.

%C 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

%H Joshua Searle and others, <a href="https://groups.google.com/g/seqfan/c/QDMpzI20qXs">Overlapping Hyperspheres</a>, discussion in SeqFan mailing list, Oct 29, 2024.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Spherical_cap#Hyperspherical_cap">Spherical Cap</a>

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>

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

%F Equals A102519/3 = A258147/2 = A358981/Pi. - _Hugo Pfoertner_, Oct 31 2024

%e 0.1955011094778853209555017087550909729839867132416731701334918282605757574660158846...

%t RealDigits[1/3 - Sqrt[3]/(4*Pi), 10, 120][[1]] (* _Amiram Eldar_, Oct 31 2024 *)

%o (Python)

%o from sympy import S, betainc_regularized, hyper

%o f = lambda n: betainc_regularized(S(n-1)/2, S(1)/2, 0, S(3)/4).rewrite(hyper).gammasimp().simplify().expand()/2

%o f(4).n(99) # decimal expansion

%o [int(d) for d in str(_)[2:-1]] # digits, omitting "0." and last (could be rounded)

%o # _M. F. Hasler_, Oct 31 2024

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

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

%K nonn,cons,easy

%O 0,2

%A _Joshua Searle_, Oct 30 2024