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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 (list; constant; graph; refs; listen; history; text; internal format)
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-Fran├žois Alcover, Jul 09 2014

STATUS

approved

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Last modified September 26 19:57 EDT 2021. Contains 347672 sequences. (Running on oeis4.)