A132696 -id:A132696 - cp's OEIS frontend

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

N. J. A. Sloane

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

			Pi/6 = 0.5235987755982988730771072305465838140328615665625176368291574...

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.

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

Cf. A000796, A132696.

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

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

A132702 Decimal expansion of 12/Pi.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 26 2007

From Bernard Schott, Apr 17 2022: (Start)
For any triangle ABC, (see Crux Mathematicorum):
(b+c)/A + (c+a)/B + (a+b)/C >= (12/Pi) * s,
b*c/(A*(s-a)) + c*a/(B*(s-b)) + a*b/(C*(s-c)) >= (12/Pi) * s,
where (A,B,C) are the angles (measured in radians), (a,b,c) the side lengths of this triangle and s the semiperimeter.
Equality stands iff triangle ABC is equilateral. (End)

			3.819718634...

S. Arslanagić and D. M. Milošević, Problem 1827, Crux Mathematicorum, Vol. 22, No. 1 (1996), p. 36.
Index entries for transcendental numbers.

Cf. A049541, A060294, A089491, A088538, A086201.

Digits:=100; evalf(12/Pi); # Wesley Ivan Hurt, Mar 26 2014

RealDigits[N[12/Pi,6! ]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jun 19 2009 *)

12/Pi \\ Charles R Greathouse IV, Dec 31 2011

Equals 2*A132696 = 4*A089491 = 6*A060294. -R. J. Mathar, Jul 29 2024

More terms from Vladimir Joseph Stephan Orlovsky, Jun 19 2009

A132714 Decimal expansion of 24/Pi.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 31 2007

			=7.639437268410976116906420641880689377654062995541909539888032514827...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for transcendental numbers

Cf. A049541, A060294, A089491, A088538, A086201, A132696 to A132699, A132701, A132702.

RealDigits[N[24/Pi,6! ]] (* Vladimir Joseph Stephan Orlovsky, Dec 03 2009 *)

PARI
```
24/Pi \\ G. C. Greubel, Feb 20 2017
```

24/Pi = Sum_{k>=0} ( (30*k+7)*C(2*k,k)^2*(Hypergeometric2F1[1/2 - k/2, -k/2, 1, 64])/(-256)^k ). - Alexander R. Povolotsky, Dec 20 2012

Another version of this identity is: Sum[(30*k+7) * Binomial[2k,k]^2 * (Sum[Binomial[k-m,m] * Binomial[k,m] * 16^m, {m,0,k/2}])/(256)^k, {k,0,infinity}]. - Alexander R. Povolotsky, Jan 25 2013

More terms from Vladimir Joseph Stephan Orlovsky, Dec 03 2009

A132721 Decimal expansion of 31/Pi.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 31 2007

			9.867606471697510817670793...

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Index entries for transcendental numbers

Cf. A049541, A060294, A089491, A088538, A086201, A132696, A132697, A132698, A132699, A132701, A132702.

RealDigits[31/Pi,10,120][[1]] (* Harvey P. Dale, Dec 28 2021 *)

31/Pi \\ Charles R Greathouse IV, Oct 01 2022

Terms a(32) and beyond from Andrew Howroyd, Jan 03 2020

A132717 Decimal expansion of 27/Pi.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 31 2007

			8.59436692696234813151972322211577554986082086998464...

Index entries for transcendental numbers.

Cf. A049541, A060294, A089491, A088538, A086201, A132696, A132697, A132698, A132699, A132701, A132702.

RealDigits[27/Pi, 10, 120][[1]] (* Amiram Eldar, Jun 13 2023 *)

27/Pi \\ Charles R Greathouse IV, Oct 01 2022

A132703 Decimal expansion of 13/Pi.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 31 2007

			4.138028520389278729990977847685373412895950789251867667439350945531316738....

Index entries for transcendental numbers

Cf. A049541, A060294, A089491, A088538, A086201, A132696-A132699, A132701, A132702.

Digits:=100; evalf(13/Pi); # Wesley Ivan Hurt, Mar 26 2014

RealDigits[N[13/Pi,6! ]] (* Vladimir Joseph Stephan Orlovsky, Dec 02 2009 *)

13/Pi \\ Charles R Greathouse IV, Oct 01 2022

More terms from Vladimir Joseph Stephan Orlovsky, Dec 02 2009

A132704 Decimal expansion of 14/Pi.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 31 2007

			4.45633840657306940152874537443040213696487008073278056493468563364911033....

Index entries for transcendental numbers

Cf. A049541, A060294, A089491, A088538, A086201, A132696-A132699, A132701, A132702.

Digits:=100; evalf(14/Pi); # Wesley Ivan Hurt, Mar 26 2014

RealDigits[N[14/Pi,6! ]] (* Vladimir Joseph Stephan Orlovsky, Dec 02 2009 *)

14/Pi \\ Charles R Greathouse IV, Oct 01 2022

More terms from Vladimir Joseph Stephan Orlovsky, Dec 02 2009

More terms from Harvey P. Dale, Oct 03 2012

A132706 Decimal expansion of 16/Pi.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 31 2007

			5.092958178940650744604280427920459585102708663694606359925355....

Bruce C. Berndt, Ramanujan’s Notebooks, Part II, Springer-Verlag, New York, 1989.

Index entries for transcendental numbers

Cf. A019683, A049541, A060294, A089491, A088538, A086201, A132696 to A132699, A132701, A132702.

RealDigits[N[16/Pi,6! ]] (* Vladimir Joseph Stephan Orlovsky, Dec 02 2009 *)

16/Pi \\ Charles R Greathouse IV, Oct 01 2022

Equals 4 + Sum_{k>=0} binomial(2*k,k)^2/((k+1)^2*16^k). - Amiram Eldar, May 21 2021

16/Pi = 5 + 1^2/(10 + 3^2/(10 + 5^2/(10 + ...))). See Berndt, Entry 25, p. 140, with n = 0 and x = 5. - Peter Bala, Feb 18 2024

More terms from Vladimir Joseph Stephan Orlovsky, Dec 02 2009

A132707 Decimal expansion of 17/Pi.

Original entry on oeis.org

5, 4, 1, 1, 2, 6, 8, 0, 6, 5, 1, 2, 4, 4, 4, 1, 4, 1, 6, 1, 4, 2, 0, 4, 7, 9, 5, 4, 6, 6, 5

Offset: 1

Omar E. Pol, Aug 31 2007

			5.4112680651244414161420479546654883091716279551755192574206896980024911195637....

Index entries for transcendental numbers

Cf. A049541, A060294, A089491, A088538, A086201, A132696 to A132699, A132701, A132702.

RealDigits[N[17/Pi,6! ]] (* Vladimir Joseph Stephan Orlovsky, Dec 02 2009 *)

17/Pi \\ Charles R Greathouse IV, Oct 01 2022

A132709 Decimal expansion of 19/Pi.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 31 2007

			6.04788783749202275921758300815554575730946653813734505241135907423807831010....

Index entries for transcendental numbers

Cf. A049541, A060294, A089491, A088538, A086201, A132696-A132699, A132701, A132702.

Digits:=100; evalf(19/Pi); # Wesley Ivan Hurt, Mar 26 2014

RealDigits[N[19/Pi,6! ]] (* Vladimir Joseph Stephan Orlovsky, Dec 02 2009 *)

19/Pi \\ Charles R Greathouse IV, Oct 01 2022

More terms from Vladimir Joseph Stephan Orlovsky, Dec 02 2009

cp's OEIS Frontend

A019673 Decimal expansion of Pi/6.

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Magma

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A132702 Decimal expansion of 12/Pi.

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Maple

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A132714 Decimal expansion of 24/Pi.

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Mathematica

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A132721 Decimal expansion of 31/Pi.

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Mathematica

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A132717 Decimal expansion of 27/Pi.

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Mathematica

PARI

A132703 Decimal expansion of 13/Pi.

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Maple

Mathematica

PARI

Extensions

A132704 Decimal expansion of 14/Pi.

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