Search: a229728 -id:a229728
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A130834
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Decimal expansion of the limit of the (2/n^2)-th power of the number of distinct dimer coverings on the n X n square grid, n even, as n goes to infinity.
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+10
9
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1, 7, 9, 1, 6, 2, 2, 8, 1, 2, 0, 6, 9, 5, 9, 3, 4, 2, 4, 7, 3, 0, 5, 4, 7, 0, 8, 9, 3, 4, 2, 9, 8, 2, 4, 3, 2, 2, 6, 8, 1, 3, 4, 3, 9, 3, 1, 3, 2, 9, 5, 4, 7, 6, 7, 7, 6, 7, 5, 8, 3, 4, 7, 6, 4, 9, 9, 4, 2, 5, 0, 7, 4, 2, 3, 7, 6, 5, 7, 8, 9, 6, 0, 1, 3, 2, 2, 6
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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REFERENCES
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S. R. Finch, Mathematical Constants, Cambridge, 2003, p. 232.
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LINKS
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FORMULA
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EXAMPLE
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1.791622812069593424730547089...
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MAPLE
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evalf(exp(2*Catalan/Pi));
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MATHEMATICA
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RealDigits[Exp[(2*Catalan)/Pi], 10, 120][[1]] (* Harvey P. Dale, Jul 17 2011 *)
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PROG
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(Magma) R:=RealField(100); Exp(2*Catalan(R)/Pi(R)); // G. C. Greubel, Aug 23 2018
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A242710
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Decimal expansion of "beta", a Kneser-Mahler polynomial constant (a constant related to the asymptotic evaluation of the supremum norm of polynomials).
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+10
5
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1, 3, 8, 1, 3, 5, 6, 4, 4, 4, 5, 1, 8, 4, 9, 7, 7, 9, 3, 3, 7, 1, 4, 6, 6, 9, 5, 6, 8, 5, 0, 6, 2, 4, 1, 2, 6, 2, 8, 9, 6, 3, 7, 2, 6, 2, 2, 3, 9, 0, 7, 0, 5, 6, 0, 1, 9, 8, 7, 6, 4, 8, 4, 5, 3, 0, 0, 5, 5, 4, 9, 6, 3, 6, 3, 6, 6, 3, 6, 2, 4, 5, 4, 0, 8, 6, 3, 9, 7, 6, 7, 9, 5, 4, 4, 2, 8, 1, 1, 6
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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REFERENCES
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S. R. Finch, Mathematical Constants, Cambridge, 2003; see Section 3.10, Kneser-Mahler polynomial constants, p. 232, and Section 5.23, Monomer-dimer constants, p. 408.
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LINKS
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FORMULA
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beta = exp(G/Pi) = exp((PolyGamma(1, 4/3) - PolyGamma(1, 2/3) + 9)/(4*sqrt(3)*Pi)), where G is Gieseking's constant (cf. A143298) and PolyGamma(1,z) the first derivative of the digamma function psi(z).
Also equals exp(-Im(Li_2( 1/2 - (i*sqrt(3))/2))/Pi), where Li_2 is the dilogarithm function.
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EXAMPLE
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1.38135644451849779337146695685...
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MATHEMATICA
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Exp[(PolyGamma[1, 4/3] - PolyGamma[1, 2/3] + 9)/(4*Sqrt[3]*Pi)] // RealDigits[#, 10, 100]& // First
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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