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 1-3 of 3 results.

A019673 Decimal expansion of Pi/6.

Original entry on oeis.org

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

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Author

N. J. A. Sloane

Keywords

Comments

From Omar E. Pol, Aug 30 2007: (Start)
Pi/6 = Volume of the inscribed ellipsoid / (Volume of the cuboid (If L1>L2>L3)).
Pi/6 = Volume of the inscribed spheroid / (Volume of the cuboid (If L1>(L2=L3))).
Pi/6 = Volume of the inscribed spheroid / (Volume of the cuboid (If L1<(L2=L3))).
Pi/6 = Volume of the inscribed sphere / (Volume of the regular hexahedron (Or cube)). (End)
Pi/6 = Surface area of the inscribed sphere / (surface area of the regular hexahedron (or cube)). - Omar E. Pol, Nov 13 2007
Decimal expansion of arctan(sqrt(1/3)). - Clark Kimberling, Sep 23 2011
Also, decimal expansion of sum( k>=1, (-120+329*k+568*k^2)/(k*(1+k)*(1+2*k)*(1+4*k)*(3+4*k)*(5+4*k)) ). - Bruno Berselli, Dec 01 2013
Atomic packing factor (APF) of the simple cubic lattice filled with spheres of the same diameter (unique example among chemical elements: polonium crystal). - Stanislav Sykora, Sep 29 2014

Examples

			Pi/6 = 0.5235987755982988730771072305465838140328615665625176368291574...

References

  • Ian Stewart, Professor Stewart's Cabinet of Mathematical Curiosities, Basic Books, a member of the Perseus Books Group, NY, 2009, "A Constant Bore", pp. 49-50 & 264-266.

Links

Crossrefs

Cf. A000796, A132696.
Cf. APF's of other crystal lattices: A093825 (hcp,fcc), A247446 (diamond cubic).

Programs

Formula

From Amiram Eldar, Aug 15 2020: (Start)
Equals Integral_{x=0..oo} 1/(x^2 + 9) dx.
Equals Integral_{x=0..oo} 1/(9*x^2 + 1) dx. (End)
Pi/6 = Sum_{n >= 1} i/(n*P(n,sqrt(-3))*P(n-1,sqrt(-3))), where i = sqrt(-1) and P(n,x) denotes the n-th Legendre polynomial. The first ten terms of the series gives the approximation Pi/6 = 0.52359877559(52...) correct to 11 decimal places - Peter Bala, Mar 16 2024

A093825 Decimal expansion of Pi/(3*sqrt(2)).

Original entry on oeis.org

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

Views

Author

Eric W. Weisstein, Apr 16 2004

Keywords

Comments

Density of densest packing of equal spheres in three dimensions (achieved for example by the fcc lattice).
Atomic packing factor (APF) of the face-centered-cubic (fcc) and the hexagonal-close-packed (hcp) crystal lattices filled with spheres of the same diameter. - Stanislav Sykora, Sep 29 2014

Examples

			0.74048048969306104116931349834344894976910361489594837...

References

  • J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer, 3rd. ed., 1998. See p. 15, line n = 3.
  • Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.7, p. 506.
  • Clifford A. Pickover, The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics (2009), at p. 126.
  • David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, p. 29.

Links

Crossrefs

Cf. A093824.
Cf. APF's of other crystal lattices: A019673 (simple cubic), A247446 (diamond cubic).
Cf. A161686 (continued fraction).
Related constants: A020769, A020789, A093766, A222066, A222067, A222068, A222069, A222070, A222071, A222072, A260646.

Programs

  • Mathematica
    RealDigits[Pi/(3 Sqrt[2]), 10, 120][[1]] (* Harvey P. Dale, Feb 03 2012 *)
  • PARI
    default(realprecision, 20080); x=10*Pi*sqrt(2)/6; for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b093825.txt", n, " ", d)); \\ Harry J. Smith, Jun 18 2009
  • PARI
    Pi/sqrt(18) \\ Charles R Greathouse IV, May 11 2017

Formula

Equals A019670*A010503. - R. J. Mathar, Feb 05 2009
Equals Integral_{x >= 0} (4*x^2 + 1)/((2*x^2 + 1)*(8*x^2 + 1)) dx. - Peter Bala, Feb 12 2025

Extensions

Entry revised by N. J. A. Sloane, Feb 10 2013

A268508 Decimal expansion of Pi*sqrt(3)/8.

Original entry on oeis.org

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

Views

Author

Stanislav Sykora, Apr 16 2016

Keywords

Comments

Atomic packing factor (APF) for the body-centered cubic lattice (bcc), one of the very common crystallographic lattice types of chemical elements and compounds.

Examples

			0.68017476158783169397277934661580839960652484303477771583870685077...

Links

Crossrefs

Cf. A000796 (Pi), A002194 (sqrt(3)), A180317 (sqrt(3)/80).
APF's of other lattices: A093825 (fcc,hcp), A019673 (simple cubic), A247446 (diamond cubic).

Programs

  • Mathematica
    RealDigits[Pi Sqrt[3]/8, 10, 120][[1]] (* Eric W. Weisstein, Jan 04 2019 *)
  • PARI
    Pi*sqrt(3)/8
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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