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Decimal expansion of Pi/(3*sqrt(2)).
21

%I #66 Aug 21 2024 17:11:08

%S 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,

%T 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,

%U 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

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

%C Density of densest packing of equal spheres in three dimensions (achieved for example by the fcc lattice).

%C 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

%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer, 3rd. ed., 1998. See p. 15, line n = 3.

%D Clifford A. Pickover, The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics (2009), at p. 126.

%D David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, p. 29.

%H Harry J. Smith, <a href="/A093825/b093825.txt">Table of n, a(n) for n = 0..20000</a>

%H James Grime and Brady Haran, <a href="https://www.youtube.com/watch?v=CROeIGfr3gs">The Best Way to Pack Spheres</a>, Numberphile video (2018).

%H J. H. Conway and N. J. A. Sloane, <a href="https://doi.org/10.1007/BF02574051">What are all the best sphere packings in low dimensions?</a>, Discr. Comp. Geom., 13 (1995), 383-403.

%H Thomas C. Hales, <a href="https://publicationsthomashales.wordpress.com/books/#post-113">Dense Sphere Packings</a>, Cambridge University Press, 2012.

%H G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/D3.html">Home page for fcc lattice</a>.

%H N. J. A. Sloane and Andrey Zabolotskiy, <a href="/A093825/a093825_1.txt">Table of maximal density of a packing of equal spheres in n-dimensional Euclidean space</a> (some values are only conjectural).

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CubicClosePacking.html">Cubic Close Packing</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EllipsoidPacking.html">Ellipsoid Packing</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SpherePacking.html">Sphere Packing</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Atomic_packing_factor">Atomic packing factor</a>.

%H <a href="/index/Fa#fcc">Index entries for sequences related to f.c.c. lattice</a>.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.

%F Equals A019670*A010503. - _R. J. Mathar_, Feb 05 2009

%e 0.74048048969306104116931349834344894976910361489594837...

%t RealDigits[Pi/(3 Sqrt[2]), 10, 120][[1]] (* _Harvey P. Dale_, Feb 03 2012 *)

%o (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

%o (PARI) Pi/sqrt(18) \\ _Charles R Greathouse IV_, May 11 2017

%Y Cf. A093824.

%Y Cf. APF's of other crystal lattices: A019673 (simple cubic), A247446 (diamond cubic).

%Y Cf. A161686 (continued fraction).

%Y Related constants: A020769, A020789, A093766, A222066, A222067, A222068, A222069, A222070, A222071, A222072, A260646.

%K nonn,cons,easy

%O 0,1

%A _Eric W. Weisstein_, Apr 16 2004

%E Entry revised by _N. J. A. Sloane_, Feb 10 2013