A244996 -id:A244996 - cp's OEIS frontend

A143298 Decimal expansion of Gieseking's constant.

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

Offset: 1

Eric W. Weisstein, Aug 05 2008

The largest possible volume of a tetrahedron in hyperbolic space. Named by Adams (1998) after German mathematician Hugo Gieseking (1887 - 1915). - Amiram Eldar, Aug 14 2020

			1.0149416064096536250...

J. Borwein and P. Borwein, Experimental and computational mathematics: Selected writings, Perfectly Scientific Press, 2010, p. 106.
Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 233, 512.

G. C. Greubel, Table of n, a(n) for n = 1..10000
Colin C. Adams, The newest inductee in the Number Hall of Fame, Mathematics Magazine, Vol. 71, No. 5 (1998), pp. 341-349.
John Campbell, Proof of a conjecture due to Sun concerning Catalan's constant, hal-03644515 [math], 2022.
John M. Campbell, On two conjectures due to Sun, Univ. Rochester, Online J. Analytic Comb. (2023) Issue 18, Art No. 4. See p. 4.
P. J. de Doelder, On the Clausen integral Cl_2(theta) and a related integral, J. Comp. Appl. Math. 11 (1984) 325-330.
K. S. Kolbig, Chebyshev coefficients for the Clausen function Cl_2(x), J. Comp. Appl. Math. 64 (1995) 295-297.
Vincent Nguyen, On Some Series Involving Harmonic and Skew-Harmonic Numbers, arXiv:2304.11614 [math.CA], 2023.
Eric Weisstein's World of Mathematics, Gieseking's Constant.
Wikipedia, Gieseking manifold.

Cf. A091518, A242710, A244345, A244996.

sqrt(3)/6*(Psi(1,1/3)-2*Pi^2/3) ; evalf(%) ; # R. J. Mathar, Sep 23 2013

N[(9 - PolyGamma[1, 2/3] + PolyGamma[1, 4/3])/(4*Sqrt[3]), 105] // RealDigits // First

polygamma(n, x) = if (n == 0, psi(x), (-1)^(n+1)*n!*zetahurwitz(n+1, x));
sqrt(3)/6*(polygamma(1, 1/3) - 2*Pi^2/3)
(9 - polygamma(1, 2/3) + polygamma(1, 4/3))/(4*sqrt(3)) \\ Gheorghe Coserea, Sep 30 2018

clausen(n, x) = my(z = polylog(n, exp(I*x))); if (n%2, real(z), imag(z));
clausen(2, Pi/3) \\ Gheorghe Coserea, Sep 30 2018

sqrt(3)/2 * sumpos(n=1, 1/(6*n-4)^2 + 1/(6*n-5)^2 - 1/(6*n-1)^2 - 1/(6*n-2)^2) \\ Gheorghe Coserea, Sep 30 2018

Equals (9 - PolyGamma(1, 2/3) + PolyGamma(1, 4/3))/(4*sqrt(3)).
Equals Sum_{k>0} sin(k*Pi/3)/k^2; (also equals (sqrt(3)/2)*Sum_{k>=1} -1/(6k-1)^2 - 1/(6k-2)^2 + 1/(6k-4)^2 + 1/(6k-5)^2). - Jean-François Alcover, Jun 19 2016, from the book by J. & P. Borwein.
From Amiram Eldar, Aug 14 2020: (Start)
Equals Integral_{x=0..2*Pi/3} log(2*cos(x/2)).
Equals (3*sqrt(3)/4) * (1 - Sum_{k>=0} 1/(3*k + 2)^2 + Sum_{k>=1} 1/(3*k + 1)^2) = (3*sqrt(3)/4) * Sum_{k>=1} A049347(k-1)/k^2.
Equals Pi * A244996 = Pi * log(A242710). (End)
Equals A091518/2 = A244345/5. - Hugo Pfoertner, Sep 16 2024

A122722 Number of triangulations of Delta^2 x Delta^(k-1).

Original entry on oeis.org

1, 6, 108, 4488, 376200, 58652640, 16119956160, 7519632382080, 5788821019685760, 7197150396467808000, 14206044114169232371200, 43903287397136367836697600, 210012592354755890839147008000, 1540026232221309103088828327116800, 17170286302440610680613970557956096000, 289015112280462271460535463614055526400000

Offset: 1

Jonathan Vos Post, Oct 22 2006

nonn

The number of triangulations of Delta^2 x Delta^(k) is between alpha^(k^2) and beta*(k^2) where alpha = (27/16)^(1/4) ~ 1.13975 and beta = 6^(1/6) ~ 1.34800 [p. 10 of Santos's handwritten notes about "The Cayley trick"].
There are arithmetic errors in Santos's lecture notes "The Cayley trick". The same table gives lozenge tilings of k*Delta^2.
From Petros Hadjicostas, Sep 13 2019: (Start)
The first column (indexed by k) of the table on p. 9 in Santos' handwritten notes "The Cayley trick" is actually the sequence (A273464(k, k*(k-1)/2 + 1): k >= 1).
In later published papers, Santos (2004, 2005) mentions that the number of triangulations of Delta^2 x Delta^k grows as exp(A244996*k^2/2 + o(k^2)) as k -> infinity. Notice that exp(A244996 * k^2/2) = A242710^(k^2/2). [See Theorem 1 and Theorem 4.9. Probably Theorem 1, part (2), in Santos (2004) has a typo.]
Note that alpha = (27/16)^(1/4) ~ 1.13975 < A242710^(k^2/2) ~ 1.175311 < beta = 6^(1/6) ~ 1.34800 (where alpha and beta are given on the first paragraph of these comments).
The reason the name of the sequence has "Delta^2 x Delta^(k-1)" rather than "Delta^2 x Delta^k" is because (according to Santos) the number of triangulations of Delta^2 x Delta^(k-1) equals k! times the number of lozenge tilings of k*Delta^2. (End)

			a(1) = 1 * 1! = 1.
a(2) = 3 * 2! = 6.
a(3) = 18 * 3! = 108.
a(4) = "187 * 4! = 2244" [sic]; actually 187 * 4! = 4488.
a(5) = "3135 * 5! = 188100" [sic]; actually 3135 * 5! = 376200.

J. A de Loera, Nonregular triangulations of products of simplices, Discrete Comp. Geom., 15(3) (1996), 253-264. [It may be related to this sequence.]
J. A. de Loera, J. Rambau, and Francisco Santos, MSRI Summer School on Triangulations of point sets, Applications, Structures and Algorithms.
J. A. De Loera, J. Rambau, and Francisco Santos, Further topics, in: Triangulations, vol 25 of Algor. Computat. Math. (2010), pp. 433-511.
R. J. Mathar, Lozenge tilings of the equilateral triangle, arXiv:1909.06336 [math.CO], 2019.
Francisco Santos, The Cayley trick, handwritten lecture notes; see table on p. 9.
Francisco Santos, The Cayley trick and triangulations of products of simplices, arXiv:math/0312069 [math.CO], 2004; see Theorem 1 (p. 2).
Francisco Santos, The Cayley trick and triangulations of products of simplices, Cont. Math. 374 (2005), pp. 151-177.
Benjamin Frederik Schröter, Matroidal subdivisions, Dressians and tropical Grassmannians, Ph.D. Dissertation, Technische Universität Berlin, Berlin, 2018; see Appendix on p. 111.

Cf. A000124, A011555, A011556, A045943, A242710, A244996, A273464, A326367, A326368, A326369.

Conjectures: a(n) = n! * A273464(n, n*(n+1)/2) for n >= 1; a(n) = A011555(n-1) for n >= 2. [A273464(n,k) is defined for n >= 1 and 0 <= k <= n*(n+1)/2.] - Petros Hadjicostas, Sep 12 2019

More terms (using the references) from Petros Hadjicostas, Sep 12 2019

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.

A244997 Decimal expansion of the moment derivative W_4'(0) associated with the radial probability distribution of a 4-step uniform random walk.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jul 09 2014

			0.42627839881750579092352142659616687305800676962963510754160645802652945...

Jonathan M. Borwein, Armin Straub, James Wan, and Wadim Zudilin, Densities of Short Uniform Random Walks p. 978, Canad. J. Math. 64(2012), 961-990.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 142.

Cf. A244996.

RealDigits[(7/2)*Zeta[3]/Pi^2, 10, 105] // First

W_4'(0) = (7/2)*zeta(3)/Pi^2.

W_4'(0) = integral over the square [0,Pi]x[0,Pi] of log(3+2*cos(x)+2*cos(y)+2*cos(x-y)) dx dy.

A245025 Decimal expansion of the moment derivative W_3'(2) associated with the radial probability distribution of a 3-step uniform random walk.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jul 10 2014

			2.1422044985256634680139197847019650201206458017918000691935563806464998832...

Jonathan M. Borwein, Armin Straub, James Wan, and Wadim Zudilin, Densities of Short Uniform Random Walks p. 978, Canad. J. Math. 64(2012), 961-990.
Eric Weisstein's MathWorld, Clausen's Integral

Cf. A244996.

Clausen2[x_] := Im[PolyLog[2, Exp[x*I]]]; RealDigits[2 + (3/Pi)*Clausen2[Pi/3] - 3*Sqrt[3]/(2*Pi), 10, 105] // First

2 + 3*imag(polylog(2, exp(Pi*I/3)))/Pi - 3*sqrt(3)/2/Pi \\ Charles R Greathouse IV, Aug 27 2014

W_3'(2) = 2 + (3/Pi)*Cl2(Pi/3) - 3*sqrt(3)/(2*Pi), where Cl2 is the Clausen function.

W_3'(2) = 2 + 3*W_3'(0) - 3*sqrt(3)/(2*Pi).

A244999 Decimal expansion of the moment derivative W_5'(0) associated with the radial probability distribution of a 5-step uniform random walk.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jul 09 2014

			0.54441256175218558519587806274502767666605280202852744228702849390214369...

Jonathan M. Borwein, Armin Straub, James Wan, and Wadim Zudilin, Densities of Short Uniform Random Walks p. 978, Canad. J. Math. 64(2012), 961-990.

Cf. A244996, A244997.

digits = 92; Log[2] - EulerGamma - NIntegrate[(BesselJ[0, x]^5 - 1)/x, {x, 0, 1}, WorkingPrecision -> digits + 20] - NIntegrate[BesselJ[0, x]^5/x, {x, 1, Infinity}, WorkingPrecision -> digits + 20] // RealDigits[#, 10, digits] & // First

W_5'(0) = log(2) - gamma - integral_{x=0..1}((J_0(x)^5-1)/x) - integral_{x>1}(J_0(x)^5/x), where J_0 is the Bessel function of the first kind.

A247447 Decimal expansion of r_(5,1), a constant which is the residue at -4 of the distribution function of the distance travelled in a 5-step uniform random walk.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Sep 17 2014

			0.0066167302594300817140577380007496562495510327524833...

Jonathan M. Borwein, Armin Straub, James Wan, and Wadim Zudilin, Densities of Short Uniform Random Walks p. 974, Canad. J. Math. 64(2012), 961-990.

Cf. A244993, A244994, A244995, A244996, A244997, A244999, A245025, A245026.

r[5, 0] = (2*Sqrt[15]*Re[HypergeometricPFQ[{1/2, 1/2, 1/2}, {5/6, 7/6}, 125/4]])/Pi^2; r[5, 1] = 13/225*r[5, 0] - 2/(5*Pi^4*r[5, 0]); Join[{0, 0}, RealDigits[r[5, 1], 10, 101] // First]

r_(5,1) = 13/225*r_(5,0) - 2/(5*Pi^4*r_(5,0)), where r_(5,0) is A244995 (residue at -2).

r_(5,1) = 13/(1800*sqrt(5))*Gamma(1/15)*Gamma(2/15)*Gamma(4/15)*Gamma(8/15)/Pi^4 - 1/sqrt(5)*Gamma(7/15)*Gamma(11/15)*Gamma(13/15)*Gamma(14/15)/Pi^4.

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A143298 Decimal expansion of Gieseking's constant.

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A122722 Number of triangulations of Delta^2 x Delta^(k-1).

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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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A244997 Decimal expansion of the moment derivative W_4'(0) associated with the radial probability distribution of a 4-step uniform random walk.

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A245025 Decimal expansion of the moment derivative W_3'(2) associated with the radial probability distribution of a 3-step uniform random walk.

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A244999 Decimal expansion of the moment derivative W_5'(0) associated with the radial probability distribution of a 5-step uniform random walk.

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A247447 Decimal expansion of r_(5,1), a constant which is the residue at -4 of the distribution function of the distance travelled in a 5-step uniform random walk.

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