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A340826
Decimal expansion of Cl_2(Pi/5), where Cl_2 is the Clausen function of order 2.
0
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
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
KEYWORD
nonn,cons
AUTHOR
Artur Jasinski, Jan 23 2021
STATUS
approved