A130834 -id:A130834 - cp's OEIS frontend

A229728 Decimal expansion of the square of the constant A130834.

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

Offset: 1

N. J. A. Sloane, Oct 01 2013

			3.209912300728157678629749481779905158748592124251834494874586...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, p. 232.

Cf. A000796, A006752, A097469, A130834.

Cf. A007341, A212800, A334088, A335586, A340139, A340176, A340557, A340562, A341479.

RealDigits[Exp[4*Catalan/Pi], 10, 120][[1]] (* Amiram Eldar, Jun 12 2023 *)

exp(4*Catalan/Pi) \\ Charles R Greathouse IV, Jul 15 2014

From Amiram Eldar, Jun 12 2023: (Start)
Equals exp(4*G/Pi) = exp(4*A006752/A000796).
Equals A097469^4. (End)

A097469 Decimal expansion of growth constant C for dimer model on square grid.

Original entry on oeis.org

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

Offset: 1

Ralf Stephan, Sep 18 2004

			1.33851515197609676693819590201851353706435369712791131464123...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 1.8.3 and 5.23.1, pp. 63, 407.

G. C. Greubel, Table of n, a(n) for n = 1..10000
F. Ardila and R. P. Stanley, Tilings, arXiv:math/0501170 [math.CO], 2005.
P. W. Kasteleyn, The Statistics of Dimers on a Lattice, Physica, 27 (1961), 1209-1225.
J. Propp, Enumeration of matchings: problems and progress, in New Perspectives in Algebraic Combinatorics

Cf. A038758, A054344, A130834, A143233.

Cf. A000796 (Pi), A006752 (Catalan's constant).

SetDefaultRealField(RealField(100)); R:=RealField(); Exp(Catalan(R)/Pi(R)); // G. C. Greubel, Aug 25 2018

RealDigits[Exp[Catalan/Pi], 10, 100][[1]] (* G. C. Greubel, Aug 25 2018 *)

default(realprecision, 100); exp(Catalan/Pi) \\ G. C. Greubel, Aug 25 2018

Equals e^(G/Pi), with G = A006752 (Catalan's constant).
Equals exp((1/Pi^2) * Integral_{x=0..Pi/2, y=0..Pi/2} log(4*cos(x)^2 + 4*cos(y)^2) dx dy). - Vaclav Kotesovec, Jan 04 2021
Equals sqrt(A130834) = exp(A143233). - Hugo Pfoertner, Nov 18 2024

Terms a(14) onward corrected by G. C. Greubel, Aug 26 2018

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.

A242711 Decimal expansion of C_3, a constant related to sharp inequalities for the product of 3 polynomials, which was introduced by David Boyd.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, May 21 2014

			1.908145626812785672415752694888439608281...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 3.10 p. 233.

David W. Boyd, Sharp inequalities for the product of polynomials
Eric Weisstein's MathWorld, Clausen's Integral

Cf. A130834 (C_2), A242712 (C_4), A242713 (C_5), A242714 (C_6).

Clausen2[x_] := Im[PolyLog[2, Exp[x*I]]]; c[m_] := Exp[m/Pi*Clausen2[Pi - Pi/m]]; RealDigits[c[3], 10, 100] // First

exp(3*imag(polylog(2,exp(2*I*Pi/3)))/Pi) \\ Charles R Greathouse IV, Jul 14 2014

exp(3/Pi*Clausen2(Pi - Pi/3)), where Clausen2 is Clausen's Integral.

A242712 Decimal expansion of C_4, a constant related to sharp inequalities for the product of 4 polynomials, which was introduced by David Boyd.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, May 21 2014

			1.9484547889588356067031024668865755583...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 3.10 p. 233.

David W. Boyd, Sharp inequalities for the product of polynomials
Eric Weisstein's MathWorld, Clausen's Integral

Cf. A130834 (C_2), A242711 (C_3), A242713 (C_5), A242714 (C_6).

Clausen2[x_] := Im[PolyLog[2, Exp[x*I]]]; c[m_] := Exp[m/Pi*Clausen2[Pi - Pi/m]]; RealDigits[c[4], 10, 100] // First

exp(4*imag(polylog(2, exp(3*I*Pi/4)))/Pi) \\ Charles R Greathouse IV, Jul 15 2014

exp(4/Pi*Clausen2(Pi - Pi/4)), where Clausen2 is Clausen's Integral.

A242713 Decimal expansion of C_5, a constant related to sharp inequalities for the product of 5 polynomials, which was introduced by David Boyd.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, May 21 2014

			1.967044901088071888351432414582828054693451...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 3.10 p. 233.

David W. Boyd, Sharp inequalities for the product of polynomials
Eric Weisstein's MathWorld, Clausen's Integral

Cf. A130834 (C_2), A242711 (C_3), A242712 (C_4), A242714 (C_6).

Clausen2[x_] := Im[PolyLog[2, Exp[x*I]]]; c[m_] := Exp[m/Pi*Clausen2[Pi - Pi/m]]; RealDigits[c[5], 10, 100] // First

exp(5*imag(polylog(2, exp(4*I*Pi/5)))/Pi) \\ Charles R Greathouse IV, Jul 15 2014

exp(5/Pi*Clausen2(Pi - Pi/5)), where Clausen2 is Clausen's Integral.

A242714 Decimal expansion of C_6, a constant related to sharp inequalities for the product of 6 polynomials, which was introduced by David Boyd.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, May 21 2014

			1.9771268308039343866983671752539756021366...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 3.10 p. 233.

David W. Boyd, Sharp inequalities for the product of polynomials
Eric Weisstein's MathWorld, Clausen's Integral

Cf. A130834 (C_2), A242711 (C_3), A242712 (C_4), A242713 (C_5).

Clausen2[x_] := Im[PolyLog[2, Exp[x*I]]]; c[m_] := Exp[m/Pi*Clausen2[Pi - Pi/m]]; RealDigits[c[6], 10, 100] // First

exp(6*imag(polylog(2, exp(5*I*Pi/6)))/Pi) \\ Charles R Greathouse IV, Jul 15 2014

exp(6/Pi*Clausen2(Pi - Pi/6)), where Clausen2 is Clausen's Integral.

A378910 Decimal expansion of 2*G/Pi, where G = A006752.

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Dec 10 2024

			0.58312180806163756027676891293678983772813230797167...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 3.10, p. 232.

Cf. A000796, A006752, A060294, A130834, A143233, A221209, A378911.

Mathematica
```
RealDigits[2Catalan/Pi,10,100][[1]]
```

Equals log(A130834).

Equals 2*A143233.

A378911 Decimal expansion of sqrt(2)exp(2G/Pi), where G = A006752.

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Dec 10 2024

			2.5337372794858419095832896340418632916896308088420...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 3.10, p. 233.

Cf. A000796, A002193, A006752, A060294, A130834, A143233, A221209, A378910.

RealDigits[Sqrt[2]*Exp[2Catalan/Pi],10,100][[1]]

A247548 Decimal expansion of D^2, a constant associated with the "Dimer Problem" on a triangular lattice.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Sep 19 2014

			2.35652735334624880922914314763999476796439150067841679838661876063419126231...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 5.23 Monomer-dimer constants p. 408.

Cf. A130834, A242710.

digits = 20; uv = Log[6 + 2*Cos[u] + 2*Cos[v] + 2*Cos[u + v]];
SetOptions[NIntegrate, WorkingPrecision -> digits + 5];
i1 = 2*NIntegrate[uv, {u, 0, Pi/2}, {v, 0, Pi/2}];
i2 = 4*NIntegrate[uv, {u, 0, Pi/2}, {v, Pi/2, Pi}];
i3 = 2*NIntegrate[uv, {u, -Pi, -Pi/2}, {v, Pi/2, Pi}];
i4 = 2*NIntegrate[uv, {u, -Pi/2, 0}, {v, 0, Pi/2}];
i5 = 4*NIntegrate[uv, {u, -Pi/2, 0}, {v, Pi/2, Pi}];
i6 = 2*NIntegrate[uv, {u, Pi/2, Pi}, {v, Pi/2, Pi}];
D2 = Exp[(1/(8*Pi^2))*(i1 + i2 + i3 + i4 + i5 + i6)];
RealDigits[D2, 10, digits] // First

exp(1/(8*Pi^2) * intnum(u=-Pi, Pi, intnum(v=-Pi,Pi, log(6 + 2*cos(u) + 2*cos(v) + 2*cos(u+v))))) \\ Michel Marcus, Sep 19 2014

Equals exp( 1/(8*Pi^2) * Integral_{v=-Pi..Pi} Integral_{u=-Pi..Pi} log(6 + 2*cos(u) + 2*cos(v) + 2*cos(u+v)) du dv).

More terms from Michel Marcus, Sep 19 2014

cp's OEIS Frontend

A229728 Decimal expansion of the square of the constant A130834.

Views

Author

Keywords

Examples

References

Crossrefs

Programs

Mathematica

PARI

Formula

A097469 Decimal expansion of growth constant C for dimer model on square grid.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

A242711 Decimal expansion of C_3, a constant related to sharp inequalities for the product of 3 polynomials, which was introduced by David Boyd.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A242712 Decimal expansion of C_4, a constant related to sharp inequalities for the product of 4 polynomials, which was introduced by David Boyd.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A242713 Decimal expansion of C_5, a constant related to sharp inequalities for the product of 5 polynomials, which was introduced by David Boyd.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A242714 Decimal expansion of C_6, a constant related to sharp inequalities for the product of 6 polynomials, which was introduced by David Boyd.

Views

Author

Keywords

Examples

References

Links

A378911 Decimal expansion of sqrt(2)exp(2G/Pi), where G = A006752.