A143298 -id:A143298 - cp's OEIS frontend

A261026 Decimal expansion of Cl_2(3*Pi/4), where Cl_2 is the Clausen function of order 2.

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

Offset: 0

Jean-François Alcover, Aug 07 2015

			0.52388935396159384919062278559400361174371628451989439444336407484...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's MathWorld, Clausen Function
Eric Weisstein's MathWorld, Clausen's Integral
Eric Weisstein's MathWorld, Barnes G-Function
Wikipedia, Clausen function
Wikipedia, Barnes G-function

Cf. A006752 (Cl_2(Pi/2) = Catalan's constant), A143298 (Cl_2(Pi/3) = Gieseking's constant), A261024 (Cl_2(2*Pi/3)), A261025 (Cl_2(Pi/4)), A261027 (Cl_2(Pi/6)), A261028 (Cl_2(5*Pi/6)).

Cl2[x_] := (I/2)*(PolyLog[2, Exp[-I*x]] - PolyLog[2, Exp[I*x]]); RealDigits[Cl2[3*Pi/4] // Re, 10, 105] // First

Equals 2*Pi*log(G(5/8)/G(3/8)) - 2*Pi*LogGamma(3/8) + (3*Pi/4) * log(2*Pi/sqrt(2+sqrt(2))), where G is the Barnes G function.

A145438 Decimal expansion of sum_{n=1..inf} 1/(n^3*binomial(2n,n)).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Feb 08 2009

Alexander Apelblat, Tables of Integrals and Series, Harri Deutsch, (1996), 4.1.47 gives Pi*sqrt(3)*(psi(2/3)-psi(1/3))/72-Zeta(3)/3 which is negative and therefore not correct.

Comment from Mikhail Kalmykov (kalmykov.mikhail(AT)googlemail.com), Jun 01 2009: Analytical results for this sum were also given in Eq. (8) of the Kalmykov and Veretin paper. These results confirm the last comment from Alois P. Heinz.

			0.522946...

J. M. Borwein, R. Girgensohn, Evaluation of Binomial Series, CECM-02-188 (2002).
A. I. Davydychev, M. Yu. Kalmykov, Massive Feynman diagrams and inverse binomial sums, Nucl. Phys. B 699 (2004), 3-64.
M. Yu. Kalmykov and O. Veretin, Single-scale diagrams and multiple binomial sums, Phys. Lett. B 483 (2000) 315-323.
R. J. Mathar, Corrigenda to "Interesting Series involving..", arXiv:0905.0215 [math.CA]

RealDigits[ N[1/18*(Sqrt[3]* Pi*(-PolyGamma[1, 2/3] + PolyGamma[1, 4/3] + 9) - 24*Zeta[3]), 105]][[1]] (* Jean-François Alcover, Nov 08 2012, after R. J. Mathar *)

Comment from Alois P. Heinz, Feb 08 2009: Maple's answer to this is: a:= sum(1/(n^3*binomial(2*n,n)), n=1..infinity); a:= 1/2 hypergeom([1, 1, 1, 1], [2, 2, 3/2], 1/4); evalf (a, 140); .522946192133335108491185183527303540163044591743977841465941014...

Equals A019693*A143298-4*A002117/3 =2*Pi*Cl_2(Pi/3)/3-4*zeta(3)/3. [From R. J. Mathar, Feb 09 2009]

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.

A258759 Decimal expansion of Ls_3(Pi/3), the value of the 3rd basic generalized log-sine integral at Pi/3 (negated).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 09 2015

			-2.0096660811305439002623537543491645038479353700110717949908496919...

Jonathan M. Borwein, Armin Straub, Special Values of Generalized Log-sine Integrals.

Cf. A258749 (Ls_3(Pi)), A258750 (Ls_4(Pi)), A258751 (Ls_5(Pi)), A258752 (Ls_6(Pi)), A258753 (Ls_7(Pi)), A258754 (Ls_8(Pi)).

Cf. A143298 (Ls_2(Pi/3)), A258760 (Ls_4(Pi/3)), A258761 (Ls_5(Pi/3)), A258762 (Ls_6(Pi/3)), A258763 (Ls_7(Pi/3)).

RealDigits[-7*Pi^3/108, 10, 105] // First

-Integral_{0..Pi/3} log(2*sin(x/2))^2 dx = -7*Pi^3/108.

A258760 Decimal expansion of Ls_4(Pi/3), the value of the 4th basic generalized log-sine integral at Pi/3.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 09 2015

			6.00949754981888891620478870620327074059696329743956841883606392675151...

Jonathan M. Borwein, Armin Straub, Special Values of Generalized Log-sine Integrals.

Cf. A258749 (Ls_3(Pi)), A258750 (Ls_4(Pi)), A258751 (Ls_5(Pi)), A258752 (Ls_6(Pi)), A258753 (Ls_7(Pi)), A258754 (Ls_8(Pi)).

Cf. A143298 (Ls_2(Pi/3)), A258759 (Ls_3(Pi/3)), A258761 (Ls_5(Pi/3)), A258762 (Ls_6(Pi/3)), A258763 (Ls_7(Pi/3)).

RealDigits[(1/2)*Pi*Zeta[3] + (9/4)*Im[ PolyLog[4, (-1)^(1/3)] - PolyLog[4, -(-1)^(2/3)]], 10, 105] // First

-Integral_{0..Pi/3} log(2*sin(x/2))^3 dx = (1/2)*Pi*zeta(3) + (9/4)*im( PolyLog(4, (-1)^(1/3)) - PolyLog(4, -(-1)^(2/3))).

Also equals 6 * 5F4(1/2,1/2,1/2,1/2,1/2; 3/2,3/2,3/2,3/2; 1/4) (with 5F4 the hypergeometric function).

A258761 Decimal expansion of Ls_5(Pi/3), the value of the 5th basic generalized log-sine integral at Pi/3 (negated).

Original entry on oeis.org

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

Offset: 2

Jean-François Alcover, Jun 09 2015

			-24.01253312551691461501571396363162679502884841064631502190162...

Jonathan M. Borwein, Armin Straub, Special Values of Generalized Log-sine Integrals.

Cf. A258749 (Ls_3(Pi)), A258750 (Ls_4(Pi)), A258751 (Ls_5(Pi)), A258752 (Ls_6(Pi)), A258753 (Ls_7(Pi)), A258754 (Ls_8(Pi)).

Cf. A143298 (Ls_2(Pi/3)), A258759 (Ls_3(Pi/3)), A258760 (Ls_4(Pi/3)), A258762 (Ls_6(Pi/3)), A258763 (Ls_7(Pi/3)).

RealDigits[-24*HypergeometricPFQ[Table[1/2, {6}], Table[3/2, {5}], 1/4], 10, 103] // First

-Integral_{0..Pi/3} log(2*sin(x/2))^4 dx = -1543*Pi^5/19440 + 6*Gl_{4, 1}(Pi/3), where Gl is the multiple Glaisher function.

Also equals -24 * 6F5(1/2,1/2,1/2,1/2,1/2,1/2; 3/2,3/2,3/2,3/2,3/2; 1/4) (with 6F5 the hypergeometric function).

A258762 Decimal expansion of Ls_6(Pi/3), the value of the 6th basic generalized log-sine integral at Pi/3.

Original entry on oeis.org

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

Offset: 3

Jean-François Alcover, Jun 09 2015

			120.0207613710553001755048886391927614834489250443014689821689519463 ...

Jonathan M. Borwein, Armin Straub, Special Values of Generalized Log-sine Integrals.

Cf. A258749 (Ls_3(Pi)), A258750 (Ls_4(Pi)), A258751 (Ls_5(Pi)), A258752 (Ls_6(Pi)), A258753 (Ls_7(Pi)), A258754 (Ls_8(Pi)).

Cf. A143298 (Ls_2(Pi/3)), A258759 (Ls_3(Pi/3)), A258760 (Ls_4(Pi/3)), A258761 (Ls_5(Pi/3)), A258763 (Ls_7(Pi/3)).

RealDigits[120* HypergeometricPFQ[Table[1/2, {7}], Table[3/2, {6}], 1/4], 10, 103] // First

-Integral_{0..Pi/3} log(2*sin(x/2))^5 dx = (15/2)*Pi*zeta(5) + (35/36)*Pi^3*zeta(3) - (135/4)*Im(-PolyLog(6, (-1)^(1/3)) + PolyLog(6, -(-1)^(2/3))).

Also equals 120 * 7F6(1/2,1/2,...; 3/2,3/2,...; 1/4) (with 7F6 the hypergeometric function).

A258763 Decimal expansion of Ls_7(Pi/3), the value of the 7th basic generalized log-sine integral at Pi/3 (negated).

Original entry on oeis.org

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

Offset: 3

Jean-François Alcover, Jun 09 2015

			-720.1245682263318010530293318351565689006935502658088138930137116778...

Jonathan M. Borwein, Armin Straub, Special Values of Generalized Log-sine Integrals.

Cf. A258749 (Ls_3(Pi)), A258750 (Ls_4(Pi)), A258751 (Ls_5(Pi)), A258752 (Ls_6(Pi)), A258753 (Ls_7(Pi)), A258754 (Ls_8(Pi)).

Cf. A143298 (Ls_2(Pi/3)), A258759 (Ls_3(Pi/3)), A258760 (Ls_4(Pi/3)), A258761 (Ls_5(Pi/3)), A258762 (Ls_6(Pi/3)).

RealDigits[-720*HypergeometricPFQ[Table[1/2, {7}], Table[3/2, {6}], 1/4], 10, 103] // First

-Integral_{0..Pi/3} log(2*sin(x/2))^5 dx = -74369*Pi^7/326592 - (15/2) * Pi * Zeta[3]^2 + 135*Gl_{6, 1}(Pi/3), where Gl is the multiple Glaisher function.

Also equals -720 * 7F6(1/2,1/2,...; 3/2,3/2,...; 1/4) (with 7F6 the hypergeometric function).

A244345 Decimal expansion of xi_3 = 5*G, the volume of an ideal hyperbolic cube, where G is Gieseking's constant.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 26 2014

			5.0747080320482681251060127713726...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 8.9 Hyperbolic Volume Constants p. 512.

Eric Weisstein's MathWorld, Gieseking's Constant
Eric Weisstein's MathWorld, Polygamma Function

Cf. A143298, A242710.

G = (9 - PolyGamma[1, 2/3] + PolyGamma[1, 4/3])/(4*Sqrt[3]); RealDigits[5*G, 10, 104] // First

5*(9 - Polygamma(1, 2/3) + Polygamma(1, 4/3)) / (4*sqrt(3)).

A384425 Decimal expansion of Sum_{k>=1} (-1)^(k+1)/(6k-5)^7 + (-1)^(k-1)/(6k-1)^7.

Original entry on oeis.org

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

Offset: 1

Jason Bard, Jun 14 2025

			1.0000115515612717521618684276082035001411926833591...

Jason Bard, Table of n, a(n) for n = 1..10000
Michael I. Shamos, A catalog of the real numbers, (2007). See p. 20 (note misprint should be "+" in the sum expression).
Index entries for transcendental numbers

Cf. A013665, A092735, A143298.

RealDigits[33367*Pi^7/100776960, 10, 1000][[1]]

Equals 33367*Pi^7/100776960.

cp's OEIS Frontend

A261026 Decimal expansion of Cl_2(3*Pi/4), where Cl_2 is the Clausen function of order 2.

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A145438 Decimal expansion of sum_{n=1..inf} 1/(n^3*binomial(2n,n)).

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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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A258759 Decimal expansion of Ls_3(Pi/3), the value of the 3rd basic generalized log-sine integral at Pi/3 (negated).

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A258760 Decimal expansion of Ls_4(Pi/3), the value of the 4th basic generalized log-sine integral at Pi/3.

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A258761 Decimal expansion of Ls_5(Pi/3), the value of the 5th basic generalized log-sine integral at Pi/3 (negated).

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A258762 Decimal expansion of Ls_6(Pi/3), the value of the 6th basic generalized log-sine integral at Pi/3.

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A258763 Decimal expansion of Ls_7(Pi/3), the value of the 7th basic generalized log-sine integral at Pi/3 (negated).

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