%I #50 Oct 05 2024 12:49:20

%S 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,

%T 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,

%U 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

%N Decimal expansion of Pi/6.

%C From _Omar E. Pol_, Aug 30 2007: (Start)

%C Pi/6 = Volume of the inscribed ellipsoid / (Volume of the cuboid (If L1>L2>L3)).

%C Pi/6 = Volume of the inscribed spheroid / (Volume of the cuboid (If L1>(L2=L3))).

%C Pi/6 = Volume of the inscribed spheroid / (Volume of the cuboid (If L1<(L2=L3))).

%C Pi/6 = Volume of the inscribed sphere / (Volume of the regular hexahedron (Or cube)). (End)

%C Pi/6 = Surface area of the inscribed sphere / (surface area of the regular hexahedron (or cube)). - _Omar E. Pol_, Nov 13 2007

%C Decimal expansion of arctan(sqrt(1/3)). - _Clark Kimberling_, Sep 23 2011

%C 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

%C 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

%D 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.

%H Ivan Panchenko, <a href="/A019673/b019673.txt">Table of n, a(n) for n = 0..1000</a>

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

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

%F From _Amiram Eldar_, Aug 15 2020: (Start)

%F Equals Integral_{x=0..oo} 1/(x^2 + 9) dx.

%F Equals Integral_{x=0..oo} 1/(9*x^2 + 1) dx. (End)

%F 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

%e Pi/6 = 0.5235987755982988730771072305465838140328615665625176368291574...

%t RealDigits[N[Pi/6,6! ]] (* _Vladimir Joseph Stephan Orlovsky_, Dec 02 2009 *)

%t RealDigits[Pi/6,10,120][[1]] (* _Harvey P. Dale_, Oct 05 2024 *)

%o (PARI) Pi/6 \\ _Charles R Greathouse IV_, Jul 07 2014

%o (Magma) C<i> := ComplexField(); [Pi(C)/6]; // _G. C. Greubel_, Nov 18 2017

%Y Cf. A000796, A132696.

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

%K nonn,cons

%O 0,1

%A _N. J. A. Sloane_