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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A060294 Decimal expansion of Buffon's constant 2/Pi.

Original entry on oeis.org

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

Views

Author

Jason Earls, Mar 28 2001

Keywords

Comments

The probability P(l,d) that a needle of length l will land on a line, given a floor with equally spaced parallel lines at a distance d (>=l) apart, is (2/Pi)*(l/d). - Benoit Cloitre, Oct 14 2002
Lim_{n->infinity} z(n)/log(n) = 2/Pi, where z(n) is the expected number of real zeros of a random polynomial of degree n with real coefficients chosen from a standard Gaussian distribution (cf. Finch reference). - Benoit Cloitre, Nov 02 2003
Also the ratio of the average chord length when two points are chosen at random on a circle of radius r to the maximum possible chord length (i.e., diameter) = A088538*r / (2*r) = 2/Pi. Is there a (direct or obvious) relationship between this fact and that 2/Pi is the "magic geometric constant" for a circle (see MathWorld link)? - Rick L. Shepherd, Jun 22 2006
Blatner (1997) says that Euler found a "fascinating infinite product" for Pi involving the prime numbers, but the number he then describes does not match Pi. Switching the numerator and the denominator results in this number. - Alonso del Arte, May 16 2012
2/Pi is also the height (the ordinate y) of the geometric centroid of each arbelos (see the references and links given under A221918) with a large radius r=1 and any small ones r1 and r2 = 1 - r1, for 0 < r1 < 1. Use the integral formula given, e.g., in the MathWorld or Wikipedia centroid reference, for the two parts of the arbelos (dissected by the vertical line x = 2*r1), and then use the decomposition formula. The heights y1 and y2 of the centroids of the two parts satisfy: F1(r1)*y1(r1) = 2*r1^2*(1-r1) and F2(1-r1)*y2(1-r1) = 2*(1-r1)^2*r1. The r1 dependent area F = F1 + F2 is Pi*r1*(1-r1). (F1 and F2 are rather complicated but their explicit formulas are not needed here.) The r1 dependent horizontal coordinate x with origin at the left tip of the arbelos is x = r1 + 1/2. - Wolfdieter Lang, Feb 28 2013
Construct a quadrilateral of maximal area inside a circle. The quadrilateral is necessarily an inscribed square (with diagonals that are diameters). 2/Pi is the ratio of the square's area to the circle's area. - Rick L. Shepherd, Aug 02 2014
The expected number of real roots of a real polynomial of degree n varies as this constant times the (natural) logarithm of n, see Kac, when its coefficients are chosen from the standard uniform distribution. This may be related to Rick Shepherd's comment. - Charles R Greathouse IV, Oct 06 2014
2/Pi is also the minimum value, at x = 1/2, on (0,1) of 1/(Pi*sqrt(x*(1-x))), the nonzero piece of the probability density function for the standard arcsine distribution. - Rick L. Shepherd, Dec 05 2016
The average distance from the center of a unit-radius circle to the midpoints of chords drawn between two points that are uniformly and independently chosen at random on the circumference of the circle. - Amiram Eldar, Sep 08 2020
2/Pi <= sin(x)/x < 1 for 0 < |x| <= Pi/2 is Jordan's inequality, also known as (2/Pi) * x <= sin(x) <= x for 0 <= x <= Pi/2; this inequality was named after the French mathematician Camille Jordan (1838-1922). - Bernard Schott, Jan 07 2023
This constant 2/Pi was named after the needle experiment, described in 1777 by the French naturalist and mathematician Georges-Louis Leclerc, Comte de Buffon (1707-1788). Note that the parrot Buffon's macaw and the antelope Buffon's kob were named also after Buffon. - Bernard Schott, Jan 10 2023
2*n*log(n)/Pi is also the dominant term in the asymptotic expansion of Sum_{k=1..n-1} csc(Pi*k/n) at n tending to infinity. - Iaroslav V. Blagouchine, Apr 21 2025

Examples

			2/Pi = 0.6366197723675813430755350534900574481378385829618257949906...

References

  • David Blatner, The Joy of Pi. New York: Walker & Company (1997): 119, circle by upper right corner.
  • G. Buffon, Essai d'arithmétique morale. Supplément à l'Histoire Naturelle, Vol. 4, 1777.
  • Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, pp. 141, 539.
  • Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, p. 196.
  • G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publ., Providence, RI, 2002, p. 7, eq. (1.2) and p. 105 eq. (7.4.2) with s=1/2.
  • Robert Kanigel, The Man Who Knew Infinity: A Life of the Genius Ramanujan, 1991.
  • Daniel A. Klain and Gian-Carlo Rota, Introduction to Geometric Probability, Cambridge, 1997, see Chap. 1.
  • Luis A. Santaló, Integral Geometry and Geometric Probability, Addison-Wesley, 1976.
  • David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 53.
  • Robert M. Young, Excursions in Calculus, An Interplay of the Continuous and the Discrete. Dolciani Mathematical Expositions Number 13. MAA.

Links

Crossrefs

Cf. A000796 (Pi), A088538, A154956, A082542 (numerators in an infinite product), A053300 (continued fraction without the initial 0).
Cf. A076668 (sqrt(2/Pi)).

Programs

  • Magma
    R:= RealField(100); 2/Pi(R); // G. C. Greubel, Mar 09 2018
  • Maple
    Digits:=100: evalf(2/Pi); # Wesley Ivan Hurt, Aug 02 2014
  • Mathematica
    RealDigits[ N[ 2/Pi, 111]][[1]]
  • PARI
    default(realprecision, 20080); x=20/Pi; for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b060294.txt", n, " ", d)); \\ Harry J. Smith, Jul 03 2009

Formula

2/Pi = 1 - 5*(1/2)^3 + 9*((1*3)/(2*4))^3 - 13*((1*3*5)/(2*4*6))^3 ... - Jason Earls [formula corrected by Paul D. Hanna, Mar 23 2013]
The preceding formula is 2/Pi = Sum_{n>=0} (-1)^n * (4*n+1) * Product_{k=1..n} (2*k-1)^3/(2*k)^3. - Alexander R. Povolotsky, Mar 24 2013. [See the Hardy reference. - Wolfdieter Lang, Nov 13 2016]
2/Pi = Product_{n>=2} (p(n) + 2 - (p(n) mod 4))/p(n), where p(n) is the n-th prime. - Alonso del Arte, May 16 2012
2/Pi = Sum_{k>=0} ((2*k)!/(k!)^2)^3*((42*k+5)/(2^{12*k+3})) (due to Ramanujan). - L. Edson Jeffery, Mar 23 2013
Equals sinc(Pi/2). - Peter Luschny, Oct 04 2019
From A.H.M. Smeets, Apr 11 2020: (Start)
Equals Product_{i > 0} cos(Pi/2^(i+1)).
Equals Product_{i > 0} f_i(2)/2, where f_0(2) = 0, f_(i+1)(2) = sqrt(2+f_i(2)) for i >= 0; a formula by François Viète (16th century).
Note that cos(Pi/2^(i+1)) = f_i(2)/2, i >= 0. (End)
Equals lim_{n->infinity} (1/n) * Sum_{k=1..n} abs(sin(k * m)) for all nonzero integers m (conjectured). Works with cos also. - Dimitri Papadopoulos, Jul 17 2020
From Amiram Eldar, Sep 08 2020: (Start)
Equals Product_{k>=1} (1 - 1/(2*k)^2).
Equals lim_{k->oo} (2*k+1)*binomial(2*k,k)^2/2^(4*k).
Equals Sum_{k>=0} binomial(2*k,k)^2/((2*k+2)*2^(4*k)). (End)
Equals Sum_{k>=0} mu(4*k+1)/(4*k+1) (Nevanlinna, 1973). - Amiram Eldar, Dec 21 2020
Equals 1 - Sum_{n >= 1} (1/16^n) * binomial(2*n, n)^2 * 1/(2*n - 1). See Young, p. 264. - Peter Bala, Feb 17 2024
Equals binomial(0, 1/2) = binomial(0, -1/2). - Peter Luschny, Dec 05 2024
From Peter Bala, Dec 10 2024:(Start)
2/Pi = 1 - 1/(2 + 2/(1 + 6/(1 + 12/(1 + 20/(1 + ... + n*(n+1)/(1 + ...), a continued fraction representation due to Euler. See A346943.
Equals 1 - (1/2)*Sum_{n >= 0} A005566(n)*(-1/4)^n. (End)

A006283 Pierce expansion for 1 / Pi.

Original entry on oeis.org

3, 22, 118, 383, 571, 635, 70529, 375687, 399380, 575584, 699357, 1561065, 1795712, 194445473, 253745996, 3199003690, 3727084011, 6607433185, 16248462801, 172940584814, 313728984965, 796022309187, 5348508258636, 5962546521072, 97497255361780, 121347007731845
Offset: 0

Views

Author

Jeffrey Shallit

Keywords

Comments

Sequence can be produced with the unit circle: a(1) = number of diameter length arcs in circle rounded down to nearest integer (remainder arc = x_1). a(2) = number of x_1 length arcs in circle rounded down to nearest integer (remainder arc = x_2). a(3) = number of x_2 length arcs in circle rounded down to nearest integer (remainder arc = x_3). And so on ... . - Peter Woodward, Sep 08 2016

Examples

			Let x_0 = 1/Pi = 0.318309886... and a(0) = floor(1/x_0) = 3. Then set x_1 = 1 - a_0*x_0 = 0.0450703..., and a(1) = floor(1/x_1) = 22. Then x_2 = 1 - a_1*x_1 = 0.008452..., and a(2) = floor(1/x2) = 118. - _Michael B. Porter_, Sep 09 2016

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A154956 (analog for 2/Pi).

Programs

  • Mathematica
    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[1/Pi, 8!], 50] (* G. C. Greubel, Nov 13 2016 *)
  • PARI
    default(realprecision, 100000); r=Pi; for(n=1,100,s=(r/(r-floor(r))); print1(floor(r),","); r=s) \\ Benoit Cloitre [amended by Georg Fischer, Nov 20 2020]

A061233 Pierce expansion for 4 - Pi.

Original entry on oeis.org

1, 7, 112, 115, 157, 372, 432, 1340, 7034, 8396, 9200, 18846, 29558, 34050, 89754, 101768, 1361737, 48461857, 81164005, 145676139, 163820009, 182446527, 5021656281, 8401618827, 22255558907, 28334352230, 127113921970, 310272097461, 782301280193, 5560255100022, 9925600136870, 85169484256928, 2542699818508737, 3145584963639199, 397021758001902006, 467746771316089905
Offset: 0

Views

Author

Frank Ellermann, May 15 2001

Keywords

Comments

Also, alternating Engel expansion for Pi.
Pi = 4 - 1/1 + 1/(1*7) - 1/(1*7*112) + 1/(1*7*112*115) - ...
Pierce expansions are always strictly increasing.

Links

Crossrefs

A014014 and A015884 are inferior versions of this sequence.
Cf. A154956 (analog for 2/Pi).

Programs

  • Maple
    Digits := 1000: x0 := 4-Pi-4^(-1000): x1 := 4-Pi+4^(-1000): ss := []: # when expansions of x0 and x1 differ, halt
k0 := floor(1/x0): k1 := floor(1/x1): while k0=k1 do ss := [op(ss),k0]: x0 := 1-k0*x0: x1 := 1-k1*x1: k0 := floor(1/x0): k1 := floor(1/x1): od:
  • Mathematica
    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[4 - Pi, 7!], 25] (* G. C. Greubel, Dec 31 2016 *)
  • PARI
    A061233(N=199)={localprec(N); my(c=4-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]} \\ optional arg fixes precision, roughly equal to total number of digits in the result. - M. F. Hasler, Nov 24 2020

Extensions

More terms from Eric Rains (rains(AT)caltech.edu), May 31 2001

A157193 Greedy Egyptian fraction expansion for 4/Pi.

Original entry on oeis.org

1, 4, 44, 1953, 4179942, 42836179578838, 3958573977160882295479936105, 36328295343356352083453782833218820307659379901717630389
Offset: 0

Views

Author

Jaume Oliver Lafont, Feb 24 2009

Keywords

Comments

Sum_{n>=0}1/a(n)=4/Pi
Truncating the series to three terms yields the convergent 22/7 as an approximation to Pi:
1+1/4+1/44=14/11=4/(22/7)

Links

Crossrefs

Cf. A088538, A154956, A156618.

Programs

  • PARI
    x=4/Pi; for (k=0,7,d=ceil(1/x);x=x-1/d;print(d,", "))
Showing 1-4 of 4 results.

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