A395468 - OEIS

%I #9 Apr 25 2026 10:47:08

%S 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,

%T 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,

%U 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

%N 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.

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

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

%C See A395465 for details, references and more links.

%H Amiram Eldar, <a href="/A395468/a395468.png">Illustration</a>.

%H Mikhail Mikhailovich Postnikov, <a href="https://www.jstor.org/stable/2589121">The Problem of Squarable Lunes</a>, The American Mathematical Monthly, Vol. 107, No. 7 (2000), pp. 645-651. Translated from Russian by Abe Shenitzer.

%H Brian J. Shelburne, <a href="https://bshelburne.wittenberguniversity.org/TheFiveLunes120408.pdf">The Five Quadrable (Squarable) Lunes</a>, Wittenberg University Springfield, 2005.

%H <a href="/index/Al#algebraic_08">Index entries for algebraic numbers, degree 8</a>.

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

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

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

%e 0.180541899848636287062369461563015127202314789707864...

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

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

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

%K nonn,cons

%O 0,2

%A _Amiram Eldar_, Apr 24 2026