login
A244996
Decimal expansion of the moment derivative W_3'(0) associated with the radial probability distribution of a 3-step uniform random walk.
7
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
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
KEYWORD
nonn,cons,walk
AUTHOR
STATUS
approved