A340826 Decimal expansion of Cl_2(Pi/5), where Cl_2 is the Clausen function of order 2.

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

Offset

1,1

Links

Table of n, a(n) for n=1..105.

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

Cf. A006752 (Cl_2(Pi/2) = Catalan's constant), A143298 (Cl_2(Pi/3) = Gieseking's constant), A261025 (Cl_2(Pi/4)), A261026 (Cl_2(3*Pi/4)), A261027 (Cl_2(Pi/6)), A261028 (Cl_2(5*Pi/6)), A340628, A340629.

Sequence in context: A011344 A257959 A137197 * A144981 A215267 A248322

Adjacent sequences: A340823 A340824 A340825 * A340827 A340828 A340829

Keyword

nonn,cons

Author

Artur Jasinski, Jan 23 2021

Status

approved