%I #39 Sep 14 2019 21:22:23

%S 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,

%T 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,

%U 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

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

%C 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

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

%H Jonathan M. Borwein, Armin Straub, James Wan, and Wadim Zudilin, <a href="http://dx.doi.org/10.4153/CJM-2011-079-2">Densities of Short Uniform Random Walks</a>, Canad. J. Math. 64(1) (2012), 961-990; see p. 978.

%H Henry Cohn, Richard Kenyon, and James Propp, <a href="https://arxiv.org/abs/math/0008220">A variational principle for domino tilings</a>, arXiv:math/0008220 [math.CO], 2000.

%H Henry Cohn, Richard Kenyon, and James Propp, <a href="https://doi.org/10.1090/S0894-0347-00-00355-6">A variational principle for domino tilings</a>, J. Amer. Math. Soc. 14(2) (2000), 297-346.

%H Francisco Santos, <a href="https://arxiv.org/abs/math/0312069">The Cayley trick and triangulations of products of simplices</a>, 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).

%H Francisco Santos, <a href="http://dx.doi.org/10.1090/conm/374/06904">The Cayley trick and triangulations of products of simplices</a>, Cont. Math. 374 (2005), 151-177.

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/ClausensIntegral.html">Clausen's Integral</a>.

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/LobachevskysFunction.html">Lobachevsky's Function</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lozenge">Lozenge</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Clausen_function">Clausen function</a>.

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

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

%F Also equals log(A242710).

%e 0.3230659472194505140936365107238063940722418407805870161308684703610151128...

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

%o (PARI) imag(polylog(2,exp(Pi*I/3)))/Pi \\ _Charles R Greathouse IV_, Aug 27 2014

%Y Cf. A122722, A273464, A242710.

%K nonn,cons,walk

%O 0,1

%A _Jean-François Alcover_, Jul 09 2014