A019673 Decimal expansion of Pi/6.

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

Author

N. J. A. Sloane

Keywords

• nonn
• cons

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

• Ivan Panchenko, Table of n, a(n) for n = 0..1000
• Wikipedia, Atomic packing factor
• Index entries for transcendental numbers

Crossrefs

Cf. A000796, A132696.

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

Programs

Magma

C := ComplexField(); [Pi(C)/6]; // G. C. Greubel, Nov 18 2017

Mathematica

RealDigits[N[Pi/6,6! ]] (* Vladimir Joseph Stephan Orlovsky, Dec 02 2009 *)
RealDigits[Pi/6,10,120][[1]] (* Harvey P. Dale, Oct 05 2024 *)

PARI

Pi/6 \\ Charles R Greathouse IV, Jul 07 2014

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