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

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

0,2

Comments

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

Links

Table of n, a(n) for n=0..86.

Joshua Searle and others, Overlapping Hyperspheres, discussion in SeqFan mailing list, Oct 29, 2024.

Wikipedia, Spherical Cap

Index entries for transcendental numbers

Formula

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

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

Example

0.1955011094778853209555017087550909729839867132416731701334918282605757574660158846...

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A089491 A199792 A193960 * A195696 A362000 A197838

Adjacent sequences: A377519 A377520 A377521 * A377523 A377524 A377525

Keyword

nonn,cons,easy

Author

Joshua Searle, Oct 30 2024

Status

approved