A244996 Decimal expansion of the moment derivative W_3'(0) associated with the radial probability distribution of a 3-step uniform random walk.

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

Offset

0,1

Comments

This constant is also associated with the asymptotic number of lozenge tilings; see the references by Santos (2004, 2005). It is called the "maximum asymptotic normalized entropy of lozenge tilings of a planar region". Santos (2004, 2005) mentions that is computed in Cohn et al. (2000). For discussion of lozenge tilings, see for example the references for sequences A122722 and A273464. - Petros Hadjicostas, Sep 13 2019

References

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003; see Section 3.10, Kneser-Mahler Polynomial Constants, p. 232.

Links

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

Jonathan M. Borwein, Armin Straub, James Wan, and Wadim Zudilin, Densities of Short Uniform Random Walks, Canad. J. Math. 64(1) (2012), 961-990; see p. 978.

Henry Cohn, Richard Kenyon, and James Propp, A variational principle for domino tilings, arXiv:math/0008220 [math.CO], 2000.

Henry Cohn, Richard Kenyon, and James Propp, A variational principle for domino tilings, J. Amer. Math. Soc. 14(2) (2000), 297-346.

Francisco Santos, The Cayley trick and triangulations of products of simplices, arXiv:math/0312069 [math.CO], 2004; see part (2) of Theorem 1 (p. 2, possible typo), Lemma 4.8 (p. 22), and Theorem 4.9 (p. 22).

Francisco Santos, The Cayley trick and triangulations of products of simplices, Cont. Math. 374 (2005), 151-177.

Eric Weisstein's MathWorld, Clausen's Integral.

Eric Weisstein's MathWorld, Lobachevsky's Function.

Wikipedia, Lozenge.

Wikipedia, Clausen function.

Formula

W_3'(0) = (1/Pi)*Cl2[Pi/3] = (3/(2*Pi))*Cl2[2*Pi/3], where Cl2 is the Clausen function.

W_3'(0) = integral_{y=1/6..5/6} log(2*sin(Pi*y)).

Also equals log(A242710).

Example

0.3230659472194505140936365107238063940722418407805870161308684703610151128...

Mathematica

Clausen2[x_] := Im[PolyLog[2, Exp[x*I]]]; RealDigits[(1/Pi)*Clausen2[Pi/3], 10, 105] // First

Prog

(PARI) imag(polylog(2, exp(Pi*I/3)))/Pi \\ Charles R Greathouse IV, Aug 27 2014

Crossrefs

Cf. A122722, A273464, A242710.

Sequence in context: A103491 A268932 A089306 * A086099 A048967 A324182

Adjacent sequences: A244993 A244994 A244995 * A244997 A244998 A244999

Keyword

nonn,cons,walk

Author

Jean-François Alcover, Jul 09 2014

Status

approved