A229728 -id:A229728 - cp's OEIS frontend

A130834 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.

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

Offset: 1

R. J. Mathar, Jul 18 2007

			1.791622812069593424730547089...

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 232, 407.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000
Steven R. Finch, Several Constants Arising in Statistical Mechanics, Ann. Comb. 3(2-4) (1999), 323-335.
Antonio Gracia Llorente, Infinite Product Formula Involving the Catalan's Constant, OSF Preprint, 2024.

Cf. A000796 (Pi), A006752 (Catalan), A097469, A229728.

R:=RealField(100); Exp(2*Catalan(R)/Pi(R)); // G. C. Greubel, Aug 23 2018

Maple
```
evalf(exp(2*Catalan/Pi));
```

RealDigits[Exp[(2*Catalan)/Pi],10,120][[1]] (* Harvey P. Dale, Jul 17 2011 *)

exp(2*Catalan/Pi) \\ Charles R Greathouse IV, Jul 15 2014

Equals exp(2*A006752/A000796).
Equals A097469^2. - Vaclav Kotesovec, Dec 30 2020
Equals Product_{k>=1} (((4*k-1)^3*(4*k+3))/((4*k+1)^3*(4*k-3)))^k. - Antonio Graciá Llorente, Jul 22 2024
Equals lim_{n->oo} 1/((4*n)^(2*n))*Product_{k=1..n} ((4*k - 1)^(4*k - 1))/((4*k - 3)^(4*k - 3)). - Antonio Graciá Llorente, Apr 16 2025

A218387 Decimal expansion of the spanning tree constant of the square lattice.

Original entry on oeis.org

1, 1, 6, 6, 2, 4, 3, 6, 1, 6, 1, 2, 3, 2, 7, 5, 1, 2, 0, 5, 5, 3, 5, 3, 7, 8, 2, 5, 8, 7, 3, 5, 7, 9, 6, 7, 5, 4, 5, 6, 2, 6, 4, 6, 1, 5, 9, 4, 3, 3, 4, 9, 0, 8, 1, 0, 4, 4, 0, 0, 6, 2, 7, 6, 4, 4, 6, 9, 9, 0, 5, 4, 7, 5, 2, 1, 7, 5, 5, 4, 4, 6, 9, 0, 6, 5, 0, 7, 2, 9, 7, 2, 1, 2, 5, 3, 6, 2, 3, 5, 6, 3, 5, 8, 9, 1, 2, 1, 1, 1, 1, 5, 1

Offset: 1

R. J. Mathar, Oct 27 2012

			1.16624361612327512055353782587357967545626461594...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 1.7 and 5.22.6, pp. 54, 399.
Asmus L. Schmidt, Ergodic theory of complex continued fractions, Number Theory with an Emphasis on the Markoff Spectrum, in: A. D. Pollington and W. Moran (eds.), Number Theory with an Emphasis on the Markoff Spectrum, Dekker, 1993, pp. 215-226.

G. C. Greubel, Table of n, a(n) for n = 1..10000
Anthony J. Guttmann, Spanning tree generating functions and Mahler measure, arXiv:1207.2815 [math-ph], 2012.
Sheldon Yang, Some properties of Catalan's constant G, Internat. J. Math. Ed. Sci. Tech. 23 (4) (1992) 549-556, L*(1).

Cf. A006752 (Catalan), A088538 (4/Pi), A229728, A247685.

R:= RealField(100); 4*Catalan(R)/Pi(R); // G. C. Greubel, Aug 23 2018

Maple
```
evalf(Catalan*4/Pi) ;
```

RealDigits[4*Catalan/Pi, 10, 100][[1]] (* G. C. Greubel, Aug 23 2018 *)

default(realprecision, 100); 4*Catalan/Pi \\ G. C. Greubel, Aug 23 2018

Equals the product of A006752 by A088538.
From Amiram Eldar, Jul 22 2020: (Start)
Equals 1 + Sum_{k>=1} (2*k-1)!!^2/((2*k)!!^2 * (2*k + 1)).
Equals Sum_{k>=0} binomial(2*k,k)^2/(16^k * (2*k + 1)). (End)
Equals (Sum_{n>=1} (-1)^(n+1)/(2*n - 1)^2) / (Sum_{n>=1} (-1)^(n+1)/(2*n - 1)) [Schmidt] (see Finch). - Stefano Spezia, Nov 07 2024
Equals log(A229728) = A247685/Pi. - Hugo Pfoertner, Nov 07 2024
Equals Integral_{x=0..1} EllipticK(x)/(Pi*sqrt(x)) dx. - Kritsada Moomuang, Jun 21 2025

A242710 Decimal expansion of "beta", a Kneser-Mahler polynomial constant (a constant related to the asymptotic evaluation of the supremum norm of polynomials).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, May 21 2014

			1.38135644451849779337146695685...

Steven 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.

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.
Kurt Mahler, A remark on a paper of mine on polynomials. [In this paper, j is log(beta) = A244996.]
Kurt Mahler, A remark on a paper of mine on polynomials, Illinois J. Math. 8(1) (1964), 1-4.
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, Gieseking's Constant.
Eric Weisstein's MathWorld, Lobachevsky's Function.
Wikipedia, Clausen function.

Cf. A122722, A130834, A143298, A229728, A244996.

Exp[(PolyGamma[1, 4/3] - PolyGamma[1, 2/3] + 9)/(4*Sqrt[3]*Pi)] // RealDigits[#, 10, 100]& // First

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.

A377764 Decimal expansion of (Pi/8)exp(4G/Pi), where G is the Catalan constant (A006752).

Original entry on oeis.org

1, 2, 6, 0, 5, 2, 9, 6, 1, 2, 8, 2, 9, 3, 8, 6, 4, 1, 0, 5, 5, 4, 5, 3, 6, 3, 3, 0, 1, 3, 5, 4, 0, 9, 8, 4, 2, 2, 0, 2, 6, 6, 9, 2, 3, 9, 3, 5, 1, 5, 8, 8, 7, 2, 2, 6, 1, 0, 7, 7, 6, 8, 3, 3, 7, 3, 4, 3, 3, 9, 2, 6, 0, 5, 9, 0, 0, 9, 3, 5, 1, 1, 8, 8, 6, 7, 0, 5, 0, 7

Offset: 1

Paolo Xausa, Nov 06 2024

			1.26052961282938641055453633013540984220266923935...

Michael Penn, A nice product from Ramanujan -- featuring 3 important constants!, YouTube video, 2023.

Cf. A000796, A001113, A005408, A006752, A016754, A019675, A229728.

First[RealDigits[Pi/8*Exp[4*Catalan/Pi], 10, 100]]

Equals A019675*A229728.
Equals Product_{k >= 1} (1 - 1/(2*k+1)^2)^((-1)^k*(2*k+1)) (from Ramanujan).
Equals Product_{k >= 1} (1 - 1/A016754(k))^((-1)^k*A005408(k)).

cp's OEIS Frontend

A130834 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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A218387 Decimal expansion of the spanning tree constant of the square lattice.

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A242710 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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Mathematica

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A377764 Decimal expansion of (Pi/8)*exp(4*G/Pi), where G is the Catalan constant (A006752).

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Examples

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Mathematica

Formula

A377764 Decimal expansion of (Pi/8)exp(4G/Pi), where G is the Catalan constant (A006752).