A019707 - OEIS

A019707

Decimal expansion of sqrt(Pi)/5.

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

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OFFSET

0,1

COMMENTS

With offset 1 this is the decimal expansion of 2*sqrt(Pi) = 3.544907..., which is the smallest possible perimeter index eta=P/sqrt(A) of all figures (not necessarily connected) in the Euclidean plane with a continuous boundary of length P (perimeter) enclosing a finite area A. The smallest value is attained only by a Euclidean planar disk. For example, eta=4 for squares, eta=2(sqrt(a/b)+sqrt(b/a))>=4 for aXb rectangles, and eta=4.559014... (A268604) for equilateral triangles. - Stanislav Sykora, Feb 08 2016

LINKS

Ivan Panchenko, Table of n, a(n) for n = 0..1000

Hadrian Quan, Fernando Roman, and Michole Washington, Infinite products and periodic sequences, 2014 MSRI-UP Research Reports (2014), pp. 60-90.

Index entries for transcendental numbers.

FORMULA

Equals 0.2*A002161.

Equals sqrt(4 * Pi)/10.

Equals -Gamma(-1/2)/10, where Gamma is Euler's gamma function. - Lee A. Newberg, Mar 05 2024

2*sqrt(Pi) = 10 * sqrt(Pi)/5 = Product_{k>=2} Gamma(1/2 - 1/2^k)/Gamma(1/2) (Quan et al., 2014, p. 88, eq. (11.2)). - Amiram Eldar, Jul 02 2026

EXAMPLE

0.35449077018110320545963349666822903655950989122447...

MATHEMATICA