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A143233
Decimal expansion of the dimer constant.
4
2, 9, 1, 5, 6, 0, 9, 0, 4, 0, 3, 0, 8, 1, 8, 7, 8, 0, 1, 3, 8, 3, 8, 4, 4, 5, 6, 4, 6, 8, 3, 9, 4, 9, 1, 8, 8, 6, 4, 0, 6, 6, 1, 5, 3, 9, 8, 5, 8, 3, 7, 2, 7, 0, 2, 6, 1, 0, 0, 1, 5, 6, 9, 1, 1, 1, 7, 4, 7, 6, 3, 6, 8, 8, 0, 4, 3, 8, 8, 6, 1, 7, 2, 6, 6, 2, 6, 8, 2, 4, 3, 0, 3, 1, 3, 4, 0, 5, 8, 9, 0, 8, 9, 7, 2
OFFSET
0,1
LINKS
Jesús Guillera and Jonathan Sondow, Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent, arXiv:math/0506319 [math.NT], 2005-2006; Ramanujan J., Vol. 16 (2008), pp. 247-270; see Example 5.5.
Eric Weisstein's World of Mathematics, Domino Tiling
FORMULA
Equals Catalan/Pi.
Also equals integral_{t=-Pi..Pi} arccosh(sqrt(cos(t)+3)/sqrt(2))/(4*Pi) dt. - Jean-François Alcover, May 14 2014
From Antonio Graciá Llorente, Oct 11 2024: (Start)
Equals Sum_{n>=0} (n/2^(n + 2)) * Sum_{k>=0} (-1)^(k + 1)*binomial(n, k)*log(2*k + 1), (Guillera and Sondow, 2008).
Equals Sum_{n>=1} n*(arccoth((4*n)/3) - 3*arccoth(4*n)). (End)
Equals A006752/Pi = log(A097469) = 2*A322757. - Hugo Pfoertner, Oct 11 2024
EXAMPLE
0.29156090403081878013...
MATHEMATICA
RealDigits[Catalan/Pi, 10, 100][[1]] (* G. C. Greubel, Aug 24 2018 *)
PROG
(PARI) default(realprecision, 100); Catalan/Pi \\ G. C. Greubel, Aug 24 2018
(Magma) SetDefaultRealField(RealField(100)); R:=RealField(); Catalan(R)/Pi(R); // G. C. Greubel, Aug 24 2018
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Eric W. Weisstein, Jul 31 2008
STATUS
approved