A143298 -id:A143298 - cp's OEIS frontend

A243311 Decimal expansion of the volume of a regular ideal hyperbolic 4-simplex.

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

Offset: 0

Jean-François Alcover, Jun 03 2014

			0.268895660169311225469294867227245664524994437...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 8.9 Hyperbolic volume constants, p. 512.

Cf. A019503, A143298 (Gieseking's constant is also the volume of a regular ideal hyperbolic 3-simplex).

RealDigits[10*Pi/3*ArcSin[1/3] - Pi^2/3, 10, 102] // First

10*Pi/3*asin(1/3) - Pi^2/3 \\ Stefano Spezia, Dec 24 2024

Equals 10*Pi/3*arcsin(1/3) - Pi^2/3.

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

Original entry on oeis.org

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: 0

Artur Jasinski, Jan 23 2021

			0.9237551681005353087119860297930...

For links see A261024.

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.

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 *)

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

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A243311 Decimal expansion of the volume of a regular ideal hyperbolic 4-simplex.

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