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 12 results. Next

A228824 Decimal expansion of 4*Pi/5.

Original entry on oeis.org

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

Views

Author

Omar E. Pol, Oct 03 2013

Keywords

Comments

With offset 2, decimal expansion of 8*Pi.
8*Pi is also the surface area of a sphere whose diameter equals the square root of 8, hence its radius equals the square root of 2. More generally x*Pi is also the surface area of a sphere whose diameter equals the square root of x. - Omar E. Pol, Dec 18 2013

Examples

			4*Pi/5 = 2.5132741228718345907701147066236023073577...
8*Pi = 25.132741228718345907701147066236023073577...

Links

Crossrefs

Cf. A002193, A228819.
Cf. A000796, A019692, A122952, A019694, A019669, A228719, A228721 (Pi through 7*Pi).

Programs

A384473 Decimal expansion of the middle interior angle (in degrees) in Albrecht Dürer's approximate construction of the regular pentagon.

Original entry on oeis.org

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

Views

Author

Stefano Spezia, May 30 2025

Keywords

Examples

			108.366120162561467008046935277164429896133431...

References

  • Alfred S. Posamentier and Herbert A. Hauptman, 101 great ideas for introducing key concepts in mathematics: a resource for secondary school teachers, Corwin Press, Inc., 2001. See pages 144-145.
  • Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 181-182.

Links

Crossrefs

Cf. A228719, A384474 (in radians).
Cf. A002194, A019824, A072097, A177870.
Cf. A384475, A384476, A384477, A384478.

Programs

  • Mathematica
    RealDigits[(3Pi/4-ArcSin[Sqrt[3]Sin[Pi/12]])180/Pi,10,100][[1]] (* or *)
RealDigits[(Pi+ArcTan[(3-Sqrt[3]+Sqrt[6Sqrt[3]-4])/(3-Sqrt[3]-Sqrt[6Sqrt[3]-4])])180/Pi,10,100][[1]]

Formula

Equals 135 - 180*arcsin(sqrt(3)*sin(Pi/12))/Pi.
Equals (Pi + arctan((3 - sqrt(3) + sqrt(6*sqrt(3) - 4))/(3 - sqrt(3) - sqrt(6*sqrt(3) - 4))))*180/Pi.
Equals (540 - 2*A384475 - A384477)/2.
A384475 < this constant < A384477.

A384474 Decimal expansion of the middle interior angle (in radians) in Albrecht Dürer's approximate construction of the regular pentagon.

Original entry on oeis.org

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

Views

Author

Stefano Spezia, May 30 2025

Keywords

Examples

			1.891345594448510418687173478952739199024779225...

References

  • Alfred S. Posamentier and Herbert A. Hauptman, 101 great ideas for introducing key concepts in mathematics: a resource for secondary school teachers, Corwin Press, Inc., 2001. See pages 144-145.
  • Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 181-182.

Links

Crossrefs

Cf. A228719, A384473 (in degrees).
Cf. A002194, A019824, A177870.
Cf. A384475, A384476, A384477, A384478.

Programs

  • Mathematica
    RealDigits[3Pi/4-ArcSin[Sqrt[3]Sin[Pi/12]],10,100][[1]] (* or *)
RealDigits[Pi+ArcTan[(3-Sqrt[3]+Sqrt[6Sqrt[3]-4])/(3-Sqrt[3]-Sqrt[6Sqrt[3]-4])],10,100][[1]]

Formula

Equals 3*Pi/4 - arcsin(sqrt(3)*sin(Pi/12)).
Equals Pi + arctan((3 - sqrt(3) + sqrt(6*sqrt(3) - 4))/(3 - sqrt(3) - sqrt(6*sqrt(3) - 4))).
Equals (3*Pi - 2*A384476 - A384478)/2.
A384476 < this constant < A384478.

A384475 Decimal expansion of the smallest interior angle (in degrees) in Albrecht Dürer's approximate construction of the regular pentagon.

Original entry on oeis.org

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

Views

Author

Stefano Spezia, May 30 2025

Keywords

Examples

			107.03782592159414945517598606453616977939418394...

References

  • Alfred S. Posamentier and Herbert A. Hauptman, 101 great ideas for introducing key concepts in mathematics: a resource for secondary school teachers, Corwin Press, Inc., 2001. See pages 144-145.

Links

Crossrefs

Cf. A228719, A384476 (in radians).
Cf. A384474, A384476, A384477, A384478.

Programs

  • Mathematica
    RealDigits[180(1 + ArcTan[(-10 + 10 Sqrt[3] + 2 Sqrt[-4 + 6 Sqrt[3]] +2 Sqrt[-34 + 28 Sqrt[3] + 5 Sqrt[6 (-2 + 3 Sqrt[3])] - 11 Sqrt[-4 + 6 Sqrt[3]]] -3 Sqrt[2 (8 - 2 Sqrt[3] -Sqrt[6 (-2 + 3 Sqrt[3])] + Sqrt[-4 + 6 Sqrt[3]])] + Sqrt[6 (8 - 2 Sqrt[3] - Sqrt[6 (-2 + 3 Sqrt[3])] + Sqrt[-4 + 6 Sqrt[3]])])/(2 + 2 Sqrt[3] + 2 Sqrt[6 (-2 + 3 Sqrt[3])] - 4 Sqrt[-4 + 6 Sqrt[3]] - 2 Sqrt[-34 + 28 Sqrt[3] + 5 Sqrt[6 (-2 + 3 Sqrt[3])] - 11 Sqrt[-4 + 6 Sqrt[3]]] - 3 Sqrt[2 (8 - 2 Sqrt[3] - Sqrt[6 (-2 + 3 Sqrt[3])] + Sqrt[-4 + 6 Sqrt[3]])] + Sqrt[6 (8 - 2 Sqrt[3] - Sqrt[6 (-2 + 3 Sqrt[3])] +Sqrt[-4 + 6 Sqrt[3]])])]/Pi),10,100][[1]]

Formula

Equals (540 - 2*A384473 - A384477)/2.

A384476 Decimal expansion of the smallest interior angle (in radians) in Albrecht Dürer's approximate construction of the regular pentagon.

Original entry on oeis.org

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

Views

Author

Stefano Spezia, May 30 2025

Keywords

Examples

			1.868162486508351777580347333859167703515452195...

References

  • Alfred S. Posamentier and Herbert A. Hauptman, 101 great ideas for introducing key concepts in mathematics: a resource for secondary school teachers, Corwin Press, Inc., 2001. See pages 144-145.

Links

Crossrefs

Cf. A228719, A384475 (in degrees).
Cf. A384473, A384474, A384477, A384478.

Programs

  • Mathematica
    RealDigits[Pi + ArcTan[(-10 + 10 Sqrt[3] + 2 Sqrt[-4 + 6 Sqrt[3]] +2 Sqrt[-34 + 28 Sqrt[3] + 5 Sqrt[6 (-2 + 3 Sqrt[3])] - 11 Sqrt[-4 + 6 Sqrt[3]]] -3 Sqrt[2 (8 - 2 Sqrt[3] -Sqrt[6 (-2 + 3 Sqrt[3])] + Sqrt[-4 + 6 Sqrt[3]])] + Sqrt[6 (8 - 2 Sqrt[3] - Sqrt[6 (-2 + 3 Sqrt[3])] + Sqrt[-4 + 6 Sqrt[3]])])/(2 + 2 Sqrt[3] + 2 Sqrt[6 (-2 + 3 Sqrt[3])] - 4 Sqrt[-4 + 6 Sqrt[3]] - 2 Sqrt[-34 + 28 Sqrt[3] + 5 Sqrt[6 (-2 + 3 Sqrt[3])] - 11 Sqrt[-4 + 6 Sqrt[3]]] - 3 Sqrt[2 (8 - 2 Sqrt[3] - Sqrt[6 (-2 + 3 Sqrt[3])] + Sqrt[-4 + 6 Sqrt[3]])] + Sqrt[6 (8 - 2 Sqrt[3] - Sqrt[6 (-2 + 3 Sqrt[3])] +Sqrt[-4 + 6 Sqrt[3]])])],10,100][[1]]

Formula

Equals (3*Pi - 2*A384474 - A384478)/2.

A384477 Decimal expansion of the largest interior angle (in degrees) in Albrecht Dürer's approximate construction of the regular pentagon.

Original entry on oeis.org

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

Views

Author

Stefano Spezia, May 30 2025

Keywords

Examples

			109.19210783168876707355415731659880064894477011...

References

  • Alfred S. Posamentier and Herbert A. Hauptman, 101 great ideas for introducing key concepts in mathematics: a resource for secondary school teachers, Corwin Press, Inc., 2001. See pages 144-145.

Links

Crossrefs

Cf. A228719, A384478 (in radians).
Cf. A384473, A384474, A384475, A384476.

Programs

  • Mathematica
    RealDigits[(90 (Pi + 4 ArcCsc[2/Sqrt[2 - Sqrt[2 (5 - 2 Sqrt[3] + Sqrt[-187 + 108 Sqrt[3]])]]]))/Pi,10,100][[1]]

Formula

Equals 540 - 2*(A384473 + A384475).
Equals (90*(Pi + 4*arccsc(2/sqrt(2 - sqrt(2*(5 - 2*sqrt(3) + sqrt(-187 + 108*sqrt(3))))))))/Pi.

A384478 Decimal expansion of the largest interior angle (in radians) in Albrecht Dürer's approximate construction of the regular pentagon.

Original entry on oeis.org

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

Views

Author

Stefano Spezia, May 30 2025

Keywords

Examples

			1.9057617988556553228528885242146948475110453569...

References

  • Alfred S. Posamentier and Herbert A. Hauptman, 101 great ideas for introducing key concepts in mathematics: a resource for secondary school teachers, Corwin Press, Inc., 2001. See pages 144-145.

Links

Crossrefs

Cf. A228719, A384477 (in degrees).
Cf. A384473, A384474, A384475, A384476.

Programs

  • Mathematica
    RealDigits[((Pi + 4 ArcCsc[2/Sqrt[2 - Sqrt[2 (5 - 2 Sqrt[3] + Sqrt[-187 + 108 Sqrt[3]])]]]))/2,10,100][[1]]

Formula

Equals 3*Pi - 2*(A384474 + A384476).
Equals ((Pi + 4*arccsc(2/sqrt(2 - sqrt(2*(5 - 2*sqrt(3) + sqrt(-187 + 108*sqrt(3))))))))/2.

A228721 Decimal expansion of 7*Pi.

Original entry on oeis.org

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

Views

Author

Omar E. Pol, Oct 03 2013

Keywords

Comments

7*Pi is also the surface area of a sphere whose diameter equals the square root of 7. More generally x*Pi is also the surface area of a sphere whose diameter equals the square root of x. - Omar E. Pol, Dec 22 2013

Examples

			21.99114857512855266923850368295652018938...

Links

Crossrefs

Cf. A000796, A019692, A122952, A019694, A019669, A228719 (Pi through 6*Pi).

Programs

A229939 Decimal expansion of 9*Pi/10.

Original entry on oeis.org

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

Views

Author

Omar E. Pol, Oct 06 2013

Keywords

Comments

With offset 2, decimal expansion of 9*Pi.
9*Pi is also the surface area of a sphere whose diameter equals the square root of 9. More generally x*Pi is also the surface area of a sphere whose diameter equals the square root of x. - Omar E. Pol, Dec 22 2013

Examples

			9*Pi/10 = 2.827433388230813914616379044951552595777...
9*Pi = 28.27433388230813914616379044951552595777...

Links

Crossrefs

Cf. A000796, A019692, A122952, A019694, A019669, A228719, A228721, A228824 (Pi through 8*Pi).

Programs

Formula

9*Pi = Sum_{j >= 0} j*(j - 1)*(j - 2)*(j - 3)*2^(j+1) / ((2*j + 1)*binomial(2*j, j)). - Peter Bala, Nov 21 2023

A387147 Decimal expansion of the dihedral angle, in radians, between triangular and square faces in an elongated pentagonal pyramid (Johnson solid J_9).

Original entry on oeis.org

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

Views

Author

Paolo Xausa, Aug 18 2025

Keywords

Comments

Also one of the dihedral angles, in radians, in an elongated pentagonal bipyramid, elongated pentagonal cupola, elongated pentagonal orthobicupola, elongated pentagonal gyrobicupola, elongated pentagonal orthocupolarotunda and elongated pentagonal gyrocupolarotunda (Johnson solids J_16, J_20, J_38, J_39, J_40 and J_41, respectively).

Examples

			2.2231544665792648052267123232835740164638926196505...

Links

Crossrefs

Cf. A384138 (J_9 volume).
Cf. other J_9 dihedral angles: A019669, A228719, A236367.
Cf. A010476.

Programs

  • Mathematica
    First[RealDigits[ArcCos[-Sqrt[(10 - Sqrt[20])/15]], 10, 100]] (* or *)
First[RealDigits[RankedMax[Union[PolyhedronData["J9", "DihedralAngles"]], 2], 10, 100]]
  • PARI
    acos(-sqrt((10 - sqrt(20))/15)) \\ Charles R Greathouse IV, Aug 19 2025

Formula

Equals arccos(-sqrt((10 - 2*sqrt[5])/15)) = arccos(-sqrt((10 - A010476)/15)).
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