A395468 Decimal expansion of the area of the constructible squarable lune whose circular arcs have central angles in a 5:3 ratio and common chord of unit length.

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

Offset

0,2

Comments

One of the five constructible squarable lunes. It was discovered by the Finnish mathematician Martin Johan Wallenius (1731-1773) in 1766.

Called the "concave octagon lune" by Shelburne (2005).

See A395465 for details, references and more links.

Links

Table of n, a(n) for n=0..104.

Amiram Eldar, Illustration.

Mikhail Mikhailovich Postnikov, The Problem of Squarable Lunes, The American Mathematical Monthly, Vol. 107, No. 7 (2000), pp. 645-651. Translated from Russian by Abe Shenitzer.

Brian J. Shelburne, The Five Quadrable (Squarable) Lunes, Wittenberg University Springfield, 2005.

Index entries for algebraic numbers, degree 8.

Formula

Equals sqrt(375 + 70*sqrt(15) - 2*sqrt(15*(3265 + 794*sqrt(15))))/30.

Equals f(5*theta) - f(3*theta), where theta = A395472, and f(x) = x/sin(x)^2 - cot(x)/4.

Minimal polynomial: 21600000*x^8 - 36000000*x^6 + 8132000*x^4 - 205200*x^2 - 729.

Example

0.180541899848636287062369461563015127202314789707864...

Mathematica

RealDigits[Sqrt[375 + 70*Sqrt[15] - 2*Sqrt[15*(3265 + 794*Sqrt[15])]]/30, 10, 120][[1]]

Prog

(PARI) sqrt(375 + 70*sqrt(15) - 2*sqrt(15*(3265 + 794*sqrt(15))))/30

Crossrefs

Cf. A395465, A395466, A395467, A395469, A395470, A395471, A395472.

Sequence in context: A335958 A388398 A389051 * A308739 A118253 A135001

Adjacent sequences: A395465 A395466 A395467 * A395469 A395470 A395471

Keyword

nonn,cons

Author

Amiram Eldar, Apr 24 2026

Status

approved