cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 10 results.

A258749 Decimal expansion of Ls_3(Pi), the value of the 3rd basic generalized log-sine integral at Pi (negated).

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Jun 09 2015

Keywords

Examples

			-2.5838563900249850146230262555917829335187740471570923078453781753171...

Links

Crossrefs

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

Programs

  • Magma
    R:= RealField(100); -Pi(R)^3/12; // G. C. Greubel, Aug 23 2018
  • Mathematica
    RealDigits[-Pi^3/12, 10, 103] // First
  • PARI
    -Pi^3/12 \\ G. C. Greubel, Aug 23 2018

Formula

-Integral_{0..Pi} log(2*sin(t/2))^2 dx = -Pi^3/12.
Also equals 2nd derivative of -Pi*binomial(x, x/2) at x=0.
It can be noticed that Ls_2(Pi) is 0, and that Ls_2(Pi/2) is Catalan's constant 0.915966... (A006752).

A258750 Decimal expansion of Ls_4(Pi), the value of the 4th basic generalized log-sine integral at Pi.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Jun 09 2015

Keywords

Examples

			5.6645597042446183908052136898788142322518455591949799463744298643...

Links

Crossrefs

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

Programs

  • Mathematica
    RealDigits[(3/2)*Pi*Zeta[3], 10, 105] // First

Formula

-Integral_{0..Pi} log(2*sin(t/2))^3 dx = (3/2)*Pi*zeta(3).
Also equals 3rd derivative of -Pi*binomial(x, x/2) at x=0.

A258751 Decimal expansion of Ls_5(Pi), the value of the 5th basic generalized log-sine integral at Pi (negated).

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Jun 09 2015

Keywords

Examples

			-24.22655837883478171663368704510531884635713927472260341881815179...

Links

Crossrefs

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

Programs

Formula

-Integral_{0..Pi} log(2*sin(t/2))^4 dx = -19*Pi^5/240.
Also equals 4th derivative of -Pi*binomial(x, x/2) at x=0.

A258752 Decimal expansion of Ls_6(Pi), the value of the 6th basic generalized log-sine integral at Pi.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Jun 09 2015

Keywords

Examples

			119.8852400579281896736967018589288678430302320347399435542106179...

Links

Crossrefs

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

Programs

  • Mathematica
    RealDigits[(45/2)*Pi*Zeta[5] + (5/4)*Pi^3*Zeta[3], 10, 105] // First

Formula

-Integral_{0..Pi} log(2*sin(t/2))^5 dx = (45/2)*Pi*zeta(5) + (5/4)*Pi^3*zeta(3).
Also equals 5th derivative of -Pi*binomial(x, x/2) at x=0.

A258754 Decimal expansion of Ls_8(Pi), the value of the 8th basic generalized log-sine integral at Pi.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Jun 09 2015

Keywords

Examples

			5040.03987911504516434562143833539315930537596167748200200213853916...

Links

Crossrefs

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

Programs

  • Mathematica
    RealDigits[(2835/4)*Pi*Zeta[7] + (315/8)*Pi^3*Zeta[5] + (133/32)*Pi^5*Zeta[3], 10, 105] // First
  • PARI
    -intnum(t=0,Pi,log(2*sin(t/2))^7) \\ Hugo Pfoertner, Jul 22 2020

Formula

-Integral_{t=0..Pi} log(2*sin(t/2))^7 = (2835/4)*Pi*zeta(7) + (315/8)*Pi^3*zeta(5) + (133/32)*Pi^5*zeta(3).
Also equals 7th derivative of -Pi*binomial(x, x/2) at x=0.

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

Views

Author

Jean-François Alcover, Jun 09 2015

Keywords

Examples

			-2.0096660811305439002623537543491645038479353700110717949908496919...

Links

Crossrefs

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)).

Programs

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

Formula

-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

Views

Author

Jean-François Alcover, Jun 09 2015

Keywords

Examples

			6.00949754981888891620478870620327074059696329743956841883606392675151...

Links

Crossrefs

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)).

Programs

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

Formula

-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

Views

Author

Jean-François Alcover, Jun 09 2015

Keywords

Examples

			-24.01253312551691461501571396363162679502884841064631502190162...

Links

Crossrefs

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)).

Programs

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

Formula

-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

Views

Author

Jean-François Alcover, Jun 09 2015

Keywords

Examples

			120.0207613710553001755048886391927614834489250443014689821689519463 ...

Links

Crossrefs

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)).

Programs

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

Formula

-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

Views

Author

Jean-François Alcover, Jun 09 2015

Keywords

Examples

			-720.1245682263318010530293318351565689006935502658088138930137116778...

Links

Crossrefs

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)).

Programs

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

Formula

-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).
Showing 1-10 of 10 results.

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