A397090 Decimal expansion of the mean reciprocal Euclidean distance from a point in a unit 5D hypercube to a given vertex of the hypercube.

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

Offset

0,1

Comments

Denoted by B_5(-1) in the paper by Bailey el al. (2010).

Links

Table of n, a(n) for n=0..104.

David H. Bailey, Jonathan M. Borwein, and Richard E. Crandall, Advances in the theory of box integrals, Math. Comp., Vol. 79, No. 291 (2010), pp. 1839-1866. See p. 1851.

David H. Bailey, Jonathan M. Borwein, and Richard E. Crandall, Addendum to the article "Advances in the theory of box integrals".

Formula

Equals (5/3)*(K_1 - Pi*log(2 + sqrt(3))/2) + Pi^2/48 - 5*sqrt(3)*arctan(1/sqrt(15))/2 + 5*log(phi), where phi is the golden ratio (A001622), and K_1 = A254134.

Example

0.83959702989032600319598511030307138077628236791140706...

Mathematica

Cl2[x_] := Im[PolyLog[2, Exp[I*x]]];
K1 = With[{theta = (ArcTan[(16 - 3*Sqrt[15])/11] + Pi)/3}, Cl2[theta] - Cl2[theta + Pi/3] - Cl2[theta - Pi/2] + Cl2[theta - Pi/6] - Cl2[3*theta + Pi/3] + Cl2[3*theta + 2*Pi/3] - Cl2[3*theta - 5*Pi/6] + Cl2[3*theta + 5*Pi/6] + (6*theta - 5*Pi/2)*Log[2 - Sqrt[3]]];
RealDigits[5/3*(Re[K1] - Pi/2*Log[2 + Sqrt[3]]) + Pi^2/48 - 5*Sqrt[3]/2*ArcTan[1/Sqrt[15]] + 5*Log[GoldenRatio], 10, 120][[1]]

Prog

(PARI) Cl_2(x) = imag(polylog(2, exp(I*x)));
my(phi = quadgen(5), t = (atan((16 - 3*sqrt(15))/11) + Pi)/3, K1 = Cl_2(t) - Cl_2(t + Pi/3) - Cl_2(t - Pi/2) + Cl_2(t - Pi/6) - Cl_2(3*t + Pi/3) + Cl_2(3*t + 2*Pi/3) - Cl_2(3*t - 5*Pi/6) + Cl_2(3*t + 5*Pi/6) + (6*t - 5*Pi/2)*log(2 - sqrt(3))); 5*(K1 - Pi*log(2 + sqrt(3))/2)/3 + Pi^2/48 - 5*sqrt(3)*atan(1/sqrt(15))/2 + 5*log(phi)

Crossrefs

Cf. A001622, A254134, A397089.

d-dimensional analogs: A244920 (d=2), A254968 (d=3), A254980 (d=4), this constant (d=5).

Sequence in context: A135005 A090734 A200614 * A011467 A246671 A069610

Adjacent sequences: A397084 A397088 A397089 * A397091 A397092 A397094

Keyword

nonn,cons

Author

Amiram Eldar, Jun 16 2026

Status

approved