A180434 - OEIS

A180434

Decimal expansion of constant 2 - Pi/2.

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

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OFFSET

0,1

COMMENTS

(2-Pi/2)*a^2 is the area of the loop of the right strophoid (also called the Newton strophoid) whose polar equation is r = a*cos(2*t)/cos(t) and whose Cartesian equation is x*(x^2+y^2) = a*(x^2-y^2) or y = +- x*sqrt((a-x)/(a+x)). See the curve with its loop at the Mathcurve link; the loop appears for -Pi/4 <= t <= Pi/4. - Bernard Schott, Jan 28 2020

The probability that if three real numbers independently and uniformly selected at random in the range (0, 1) can be the lengths of the sides of a triangle then this triangle is acute (Lessard, 2017) (cf. A210958 which is half this constant). - Amiram Eldar, Apr 22 2026

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Robert Ferréol, Right strophoid, Math Curve.

Nikita Kalinin and Mikhail Shkolnikov, The number Pi and summation by SL(2,Z), arXiv:1701.07584 [math.NT], 2016. Gives a formula.

Laurent Lessard, Sticks in the woods, Book Proofs, 2017.

Index entries for transcendental numbers.

FORMULA

Equals Integral_{t=0..Pi/4} ((cos(2*t))/cos(t))^2 dt. - Bernard Schott, Jan 28 2020

From Amiram Eldar, May 30 2021: (Start)

Equals Sum_{k>=1} 2^k/(binomial(2*k,k)*k*(2*k + 1)).

Equals Integral_{x=0..1} arcsin(x)*arccos(x) dx. (End)

Equals Integral_{x=0..1} sqrt(x)/(1+x) dx. - Andy Nicol, Mar 23 2024