A060294 -id:A060294 - cp's OEIS frontend

A132702 Decimal expansion of 12/Pi.

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

A132699 Decimal expansion of 9/Pi.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 26 2007

9/Pi = 2.864788975654...

Index entries for transcendental numbers

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

Digits:=100; evalf(9/Pi); # Wesley Ivan Hurt, Mar 26 2014

RealDigits[N[9/Pi,6! ]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jun 19 2009 *)

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

More terms from Vladimir Joseph Stephan Orlovsky, Jun 19 2009

A132701 Decimal expansion of 11/Pi.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 26 2007

			3.501408748021697...

Index entries for transcendental numbers

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

Digits:=100; evalf(11/Pi); # Wesley Ivan Hurt, Mar 26 2014

RealDigits[N[11/Pi,6! ]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jun 19 2009 *)

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

More terms from Vladimir Joseph Stephan Orlovsky, Jun 19 2009

A076668 Decimal expansion of sqrt(2/Pi).

Original entry on oeis.org

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

Offset: 0

Zak Seidov, Oct 25 2002

This is the limit of (n+1)!!/n!!/n^(1/2) at n_even->inf.

Expected value of |x - mu|/sigma for normal distribution with mean mu and standard deviation sigma (i.e., the normalized mean absolute deviation). - Stanislav Sykora, Jun 30 2017

			0.79788456080286535587989211986876373695171726232986931533...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Harmann König, Carsten Schütt, and Nicole Tomczak-Jaegermann, Projection constants of symmetric spaces and variants of Khintchine's inequality, J. reine angew. Math. 511 (1999), pp. 1-42.
Index entries for transcendental numbers

Cf. A004730, A004731, A019727, A060294 (Buffon's constant 2/Pi), A092678 (probable error).

pi:=Sqrt(2/Pi(RealField(110))); Reverse(Intseq(Floor(10^110*pi))); // Vincenzo Librandi, Jul 01 2017

RealDigits[Sqrt[2/Pi],10,120][[1]] (* Harvey P. Dale, Feb 05 2012 *)

sqrt(2/Pi) \\ G. C. Greubel, Sep 23 2017

Equals A087197*A002193. - R. J. Mathar Feb 05 2009

Equals integral_{-infinity..infinity} (1-erf(x)^2)/2 dx. - Jean-François Alcover, Feb 25 2015

More terms and better description from Benoit Cloitre and Michael Somos, Oct 29 2002

Leading zero removed, offset changed by R. J. Mathar, Feb 05 2009

A112628 Decimal expansion of 2*sqrt(2)/Pi.

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Jan 11 2006

Example of extension to Buffon's Needle Problem: The probability that the boundary of a square will intersect one of the parallel lines if the square's diagonal length l (almost) equals the distance d between each pair of lines. This follows directly from the Weisstein/MathWorld Buffon's Needle Problem link's statement P=p/(Pi*d), where P is the probability of intersection with any convex polygon's boundary if the generalized diameter of that polygon is less than d and p is the perimeter of the polygon. (Take d=l, then p=2*sqrt(2)*d.).

The area of a regular octagon circumscribed in a unit-area circle. - Amiram Eldar, Nov 05 2020

			0.9003163161571060695551991910067405826645741499552206255714374712314587307...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Buffon's needle problem.
Eric Weisstein's World of Mathematics, Generalized Diameter.
Index entries for transcendental numbers

Cf. A060294 (2/Pi), A089491 (3/Pi), A224268.

R:= RealField(100); 2*Sqrt(2)/Pi(R); // G. C. Greubel, Aug 17 2018

RealDigits[2 Sqrt[2]/Pi, 10, 110][[1]] (* Bruno Berselli, Apr 02 2013 *)
(* From the second comment: *) RealDigits[N[Product[1 - 1/(4 n)^2, {n, 1, Infinity}], 110]][[1]] (* Bruno Berselli, Apr 02 2013 *)

PARI
```
2*sqrt(2)/Pi
```

Equals Product_{n>=1} (1-1/(4*n)^2). - Bruno Berselli, Apr 02 2013
Equals sinc(Pi/4). - Peter Luschny, Oct 04 2019
Equals Product_{k>=3} cos(Pi/2^k). - Amiram Eldar, Aug 24 2020

A240935 Decimal expansion of 3sqrt(3)/(4Pi).

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Aug 03 2014

A triangle of maximal area inside a circle is necessarily an inscribed equilateral triangle. This constant is the ratio of the triangle's area to the circle's area. In general, the ratio of an arbitrary triangle's area to the area of its unique Steiner ellipse, which has the least area of any circumscribed ellipse (an equilateral triangle's Steiner ellipse is a circle).

Also the probability that the distance between 2 randomly selected points within a circle will be larger than the radius. - Amiram Eldar, Mar 03 2019

			0.4134966715663440371334948737347270810480...

B. F. Finkel, Problem 38, solved by O. W. Anthony, Henry Heaton, and G. B. M. Zerr, The American Mathematical Monthly, Vol. 3, No. 12 (1896), pp. 324-326.
Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), p.52
I. Todhunter, A treatise on the Integral Calculus, London and Cambridge: MacMillan and Co., 1868, page 320, Example 7.
Wikipedia, Steiner ellipse
Index entries for transcendental numbers

Cf. A073010, A060294, A003881, A002194, A000796, A102519, A086089.

Digits:=100: evalf(3*sqrt(3)/(4*Pi)); # Wesley Ivan Hurt, Aug 03 2014

Flatten[RealDigits[3 Sqrt[3]/(4 Pi), 10, 100, -1]] (* Wesley Ivan Hurt, Aug 03 2014 *)

default(realprecision, 120);
3*sqrt(3)/(4*Pi)

3*sqrt(3)/(4*Pi) = 3*A002194/(4*A000796).

Equals A093604^2. - Hugo Pfoertner, May 18 2024

A308716 Decimal expansion of 2*sinh(Pi/2)/Pi.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Jun 19 2019

			1.465052383336634877609179374112651007819934264167...

Cf. A016742, A060294, A156648, A308715.

RealDigits[2 Sinh[Pi/2]/Pi, 10, 120][[1]]

2*sinh(Pi/2)/Pi \\ Michel Marcus, Jun 20 2019

Equals Product_{k>=1} (1 + 1/(4*k^2)).
Equals Product_{k>=1} (1 + 1/A016742(k)).
Equals binomial(0, i/2), where i is the imaginary unit. - Amiram Eldar, Nov 25 2020

A091400 a(n) = Product_{ odd primes p | n } (1 + Legendre(-1,p) ).

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 0, 1, 0, 2, 0, 0, 2, 0, 0, 1, 2, 0, 0, 2, 0, 0, 0, 0, 2, 2, 0, 0, 2, 0, 0, 1, 0, 2, 0, 0, 2, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 1, 4, 0, 0, 2, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 4, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 2, 0, 0, 2, 0

Offset: 1

N. J. A. Sloane, Mar 02 2004

			G.f. = x + x^2 + x^4 + 2*x^5 + x^8 + 2*x^10 + 2*x^13 + x^16 + 2*x^17 + 2*x^20 + ...

Goro Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, 1971, see p. 25, Eq. (2) (but without the restriction that a(4k) = 0).

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A060294, A091379, A122856, A122865, A129448.

with(numtheory): A091400 := proc(n) local i,t1,t2; t1 := ifactors(n)[2]; t2 := 1; for i from 1 to nops(t1) do if t1[i][1] > 2 then t2 := t2*(1+legendre(-1,t1[i][1])); fi; od: t2; end;
with(numtheory): seq(mul(1+legendre(-1,p),p in select(isprime, divisors(n) minus {2})),n=1..105); # Peter Luschny, Apr 20 2016

Legendre[-1, p_] := Which[p==2, 0, Mod[p, 4]==1, 1, True, -1]; a[1] = 1; a[n_] := Times @@ (Legendre[-1, #] + 1&) /@ FactorInteger[n][[All, 1]]; Array[a, 105] (* Jean-François Alcover, Dec 01 2015 *)
Join[{1},Table[Product[1+JacobiSymbol[-1,p],{p,Complement[FactorInteger[n][[All, 1]], {2}]}], {n,2,105}]] (* Peter Luschny, Apr 20 2016 *)

{a(n)=if(n<1,0,sumdiv(n,d,(-1)^bigomega(d)*moebius(d)*if(d%2,(-1)^(d\2),0)))} \\ Benoit Cloitre, Apr 17 2016

Here we use the definition that Legendre(-1, 2) = 0, Legendre(-1, p) = 1 if p == 1 mod 4, = -1 if p == 3 mod 4. This is Shimura's definition, which is different from Maple's.
a(n) is multiplicative with:
   a(2^e) = 1 for e >= 0,
   a(p^e) = 0 if p == 3 (mod 4) for e > 0,
   a(p^e) = 2 if p == 1 (mod 4) for e > 0.
(corrected by Werner Schulte, Dec 12 2020).
a(2*n) = a(n). a(3*n) = a(4*n + 3) = 0.
a(3*n + 1) = A122865(n). a(3*n + 2) = A122856(n).
a(n) = Sum_{d|n} b(d)*(-1)^bigomega(d)*moebius(d) where b(2n)=0 and b(2n+1)=(-1)^n. - Benoit Cloitre, Apr 17 2016
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2/Pi = 0.636619... (A060294). - Amiram Eldar, Oct 11 2022

Definition clarified by Peter Luschny, Apr 20 2016

A154956 Pierce expansion of 2/Pi.

Original entry on oeis.org

1, 2, 3, 5, 10, 71, 868, 1788, 7455, 44266, 54626, 74153, 224166, 390471, 1489304, 3737961, 22277163, 37201631, 113275744, 165029426, 2642368758, 3362202939, 5191046363, 8438525012, 36226438506, 40174126779, 125336047846, 531802867080, 599020778171

Offset: 0

Jaume Oliver Lafont, Jan 18 2009

nonn

			1 - 1/2(1 - 1/3(1 - 1/5(1 - 1/10(1 - 1/71)))) = 2/(355/113).

Simon Plouffe and G. C. Greubel, Table of n, a(n) for n = 0..500 (terms from 0 to 216 were computed by Simon Plouffe)

Cf. A006283 (1/Pi), A061233 (4 - Pi).

Cf. A060294 (decimal expansion of 2/Pi). - R. J. Mathar, Jan 21 2009

Digits := 300: Pierce := proc(x) local resid,a,i,an ; resid := x ; a := [] ; for i from 1 do an := floor(1./resid) ; a := [op(a),an] ; resid := evalf(1.-an*resid) ; if ilog10( mul(i,i=a)) > 0.7*Digits then break ; fi ; od: RETURN(a) ; end: a060294 := evalf(2/Pi) ; Pierce(a060294) ; # R. J. Mathar, Jan 21 2009

PierceExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@ NestList[{Floor[1/Expand[1 - #[[1]] #[[2]]]], Expand[1 - #[[1]] #[[2]]]} &, {Floor[1/(A - Floor[A])], A - Floor[A]}, n - 1]]; PierceExp[N[2/Pi, 7!], 50] (* G. C. Greubel, Nov 13 2016 *)

A154956(N=99)={localprec(N); my(c=2/Pi, d=c+c/10^N, a=[1\c]); while(a[#a]==1\d&&c=1-c*a[#a], d=1-d*a[#a]; a=concat(a, 1\c)); a[^-1]}  \\ The optional argument is the precision used, approx. equal to the total number of digits in the result. - M. F. Hasler, Jul 04 2016

More terms from R. J. Mathar, Jan 21 2009

Offset in b-file corrected by N. J. A. Sloane, Aug 31 2009

A232247 Decimal expansion of the arctan of 2/Pi.

Original entry on oeis.org

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

Offset: 0

Bruno Berselli, Nov 21 2013

			0.56691150494100940508289774672261915380648023909268233575775947204589...

Cf. A060294, A232182.

Maple
```
evalf(arctan(2/Pi));
```
Mathematica
```
RealDigits[ArcTan[2/Pi], 10, 90][[1]]
```

atan(2/Pi) \\ Charles R Greathouse IV, Mar 24 2021

Equals A019669 - A232182.

Equals Sum_{k>=0} (-1)^k*(2/Pi)^(1+2*k)/(1+2*k).

cp's OEIS Frontend

A132702 Decimal expansion of 12/Pi.

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A132699 Decimal expansion of 9/Pi.

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A132701 Decimal expansion of 11/Pi.

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A076668 Decimal expansion of sqrt(2/Pi).

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Magma

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A112628 Decimal expansion of 2*sqrt(2)/Pi.

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Magma

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A240935 Decimal expansion of 3*sqrt(3)/(4*Pi).

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A308716 Decimal expansion of 2*sinh(Pi/2)/Pi.

A240935 Decimal expansion of 3sqrt(3)/(4Pi).