OFFSET
1,1
LINKS
For links see A261024.
FORMULA
A = Cl_2(Pi/5).
B = Cl_2(2*Pi/5).
C = Cl_2(3*Pi/5).
D = Cl_2(4*Pi/5).
4*(A^2 + C^2) = 5*(B^2 + D^2).
B = 2*A - 2*D.
D = 2*B - 2*C.
2*C = 4*A - 5*D.
B = -D + sqrt(A*(2*C+D)+D^2).
B^2 + D^2 = 4*Pi^4/(325*A340628^2).
B^2 + D^2 = (13/1125)*A340629^2*Pi^4.
Equals Pi*(2*log(G(9/10) / G(11/10)) + log(Pi*(1+sqrt(5)))/5), where G is the Barnes G-function. - Vaclav Kotesovec, Jan 23 2021
EXAMPLE
0.9237551681005353087119860297930...
MATHEMATICA
Cl2[x_] := (I/2)*(PolyLog[2, Exp[-I*x]] - PolyLog[2, Exp[I*x]]); RealDigits[Re[Cl2[Pi/5]], 10, 105] // First
N[Pi*(ArcCsch[2] + Log[2*Pi*BarnesG[9/10]^10 / BarnesG[11/10]^10])/5, 120] (* Vaclav Kotesovec, Jan 23 2021 *)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Artur Jasinski, Jan 23 2021
STATUS
approved