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

Author

Jean-François Alcover, Jul 09 2014

Keywords

• nonn
• cons
• walk

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

Examples

			0.3230659472194505140936365107238063940722418407805870161308684703610151128...
		

References

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

Links

• 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.

Crossrefs

Cf. A122722, A273464, A242710.

Programs

Mathematica

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

PARI

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

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).