cp's OEIS Frontend

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 1-7 of 7 results.

A019669 Decimal expansion of Pi/2.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

Comments

With offset 2, decimal expansion of 5*Pi. - Omar E. Pol, Oct 03 2013
Decimal expansion of the number of radians in a quadrant. - John W. Nicholson, Oct 07 2013
Not the same as A085679. First differing term occurs at 10^-49, as list -49, or 51st counting term (a(-49)= 5 and A085679(-49) = 4). - John W. Nicholson, Oct 07 2013
5*Pi is also the surface area of a sphere whose diameter equals the square root of 5. More generally x*Pi is also the surface area of a sphere whose diameter equals the square root of x. - Omar E. Pol, Dec 22 2013
Pi/2 is also the radius of a sphere whose surface area equals the volume of the circumscribed cube. - Omar E. Pol, Dec 27 2013

Examples

			Pi/2 = 1.570796326794896619231321691639751442098584699...
5*Pi = 15.70796326794896619231321691639751442098584699...

References

  • Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 1.4.1 and 1.4.2, pp. 20-21.

Links

Crossrefs

Cf. A053300 (continued fraction), A060294 (2/Pi).
Cf. A000796, A019692, A122952, A019694 (Pi through 4*Pi), A106854.

Programs

Formula

Pi/2 = log(i)/i, where i = sqrt(-1). - Eric Desbiaux, Jun 27 2009
Pi/2 = Product_{n>=1} (n/(n+1))^((-1)^n) = 2 * 2/3 * 4/3 * 4/5 * 6/5 * 6/7 * 8/7 * 8/9 * 10/9 * ... (Wallis formula). - William Keith and Alonso del Arte, Jun 24 2012
Equals Sum_{k>1} 2^k/binomial(2*k,k). - Bruno Berselli, Sep 11 2015
The previous result is the particular case n = 1 of the more general identity: Pi/2 = 4^(n-1) * n!/(2*n)! * Sum_{k >= 2} 2^(k+1)*(k + n - 1)!*(k + 2*n - 2)!/(2*k + 2*n - 2)! valid for n = 0,1,2,... . - Peter Bala, Oct 26 2016
Pi/2 = Product_{n>=1} (4*n^2)/(4*n^2-1). - Fred Daniel Kline, Oct 29 2016
Pi/2 = lim_{n->oo} F(2^(n+3))/2, with one half of the area of a regular 2^(n+3)-gon, for n >= 0, inscribed in the unit circle, written as iterated square roots of 2 as F(2^(n+3))/2 = 2^n*sqrt(2 + sq2(n)), with sq2(n) = sqrt(2 + sq2(n-1)), n >= 1, with input sq2(0) = 0 (2 appears n times in sq2(n)). Viète's infinite product formula works with the partial product F(2^(n+2))/2 = Product_{j=1..n} (2/sq2(j)), n >= 1, which corresponds to the above given formula. - Wolfdieter Lang, Jul 06 2018
Pi/2 = Integral_{x = 0..oo} sin(x)^2/x^2 dx = 1/2 + Sum_{n >= 1} sin(n)^2/n^2, by the Abel-Plana formula. - Peter Bala, Nov 05 2019
From Amiram Eldar, Aug 15 2020: (Start)
Equals Sum_{k>=0} k!/(2*k + 1)!!.
Equals Sum_{k>=0} (-1)^k/(k + 1/2).
Equals Integral_{x=0..oo} 1/(x^2 + 1) dx.
Equals Integral_{x=0..oo} sin(x)/x dx.
Equals Integral_{x=0..oo} exp(x/2)/(exp(x) + 1) dx.
Equals Product_{p prime > 2} p/(p + (-1)^((p-1)/2)). (End)
Pi/2 = Integral_{x = 0..oo} 1/(1 - x^2 + x^4) dx = (1 + 2/3 + 1/5) - (1/7 + 2/9 + 1/11) + (1/13 + 2/15 + 1/17) - .... - Peter Bala, Jul 22 2022
Equals arcsin(9/10) + sqrt(19)*Sum_{k >= 1} A106854(k-1)/(k*10^k) (see Bailey and Crandall, 2001). - Paolo Xausa, Jul 15 2024
Equals 2F1(1/2,1/2 ; 3/2; 1). - R. J. Mathar, Aug 20 2024
Pi/2 = [1;1,1/2,1/3,...,1/n,...] by Wallis's approximation. - Thomas Ordowski, Oct 19 2024
From Stefano Spezia, Oct 21 2024: (Start)
Equals Sum_{k>=0} 2^k/((2*k + 1)*binomial(2*k,k)) (see Finch).
Equals Limit_{n->oo} 2^(4*n)/((2*n + 1)*binomial(2*n,n)^2) (see Finch). (End)
Equals Integral_{x=-oo..oo} sech((2*x^3 + x^2 - 5*x)/(x^2 - 1)) dx. - Kritsada Moomuang, May 29 2025

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)

A096456 Numerators of convergents to Pi/2.

Original entry on oeis.org

1, 2, 3, 11, 344, 355, 51819, 52174, 260515, 573204, 4846147, 5419351, 37362253, 42781604, 122925461, 411557987, 534483448, 2549491779, 3083975227, 17969367914, 21053343141, 881156436695, 902209779836, 2685575996367
Offset: 1

Views

Author

N. J. A. Sloane, Aug 16 2004

Keywords

Examples

			1, 2, 3/2, 11/7, 344/219, 355/226, ...

Links

Crossrefs

Cf. A096463 (denominators), A053300.
Cf. A002485 (numerators of convergents to Pi).

Programs

  • Mathematica
    Table[Numerator[FromContinuedFraction[ContinuedFraction[Pi/2, n]]], {n, 1, 25}] (* Stefan Steinerberger, Mar 18 2006 *)
  • PARI
    contfracpnqn(c=contfrac(Pi/2),#c)[1,] \\ M. F. Hasler, Oct 13 2020

Extensions

More terms from Stefan Steinerberger, Mar 18 2006

A096464 Let p(k)/q(k) = A096456(k)/A096463(k) be the k-th convergent to Pi/2; sequence gives numbers n such that |tan(p(n))|/p(n) sets a new maximal record.

Original entry on oeis.org

1, 4, 118, 136, 315, 3727, 3763, 15503, 153396, 156559, 984404, 1119377
Offset: 1

Views

Author

N. J. A. Sloane, Aug 16 2004

Keywords

Examples

			The fifth term is 315. This means that at p(315), which is a number near 2.37*10^154, |tan(p(315))|/p(315) sets a new record, a number near 556.31.

Links

Crossrefs

Cf. A096463, A096456, A053300.

A096463 Denominators of convergents to Pi/2.

Original entry on oeis.org

1, 2, 7, 219, 226, 32989, 33215, 165849, 364913, 3085153, 3450066, 23785549, 27235615, 78256779, 262005952, 340262731, 1623056876, 1963319607, 11439654911, 13402974518, 560961610149, 574364584667, 1709690779483
Offset: 1

Views

Author

N. J. A. Sloane, Aug 16 2004

Keywords

Examples

			1, 2, 3/2, 11/7, 344/219, 355/226, ...

Links

Crossrefs

Cf. A096456, A053300.

Extensions

More terms from Matthew Conroy, Jan 16 2006

A076587 First occurrence of n as a term in the continued fraction for Pi/2.

Original entry on oeis.org

1, 10, 4, 9, 20, 13, 26, 11, 142, 102, 70, 93, 179, 69, 127, 283, 52, 1166, 141, 605, 100, 83, 280, 414, 451, 61, 30, 234, 848, 448, 5, 372, 1389, 2445, 2082, 498, 603, 2565, 517, 3715, 22, 1155, 419, 856, 4125, 1573, 441, 207, 42, 1536, 5359, 576, 6654, 1002
Offset: 1

Views

Author

Benoit Cloitre, Oct 20 2002

Keywords

Crossrefs

Cf. A032523, A053300.

Programs

  • Mathematica
    Module[{nn=6700,p2},p2=ContinuedFraction[Pi/2,nn];Table[Position[p2,n,1,1],{n,60}]]//Flatten (* Harvey P. Dale, Jul 14 2023 *)
  • PARI
    default(realprecision, 1500); v=contfrac(Pi/2); a(n)=if(n<0,0,s=1; while(abs(n-component(v,s))>0,s++); s)

A096731 Values of continued fraction for Pi/2 associated with the records in A096464.

Original entry on oeis.org

1, 31, 84, 116, 873, 4319, 16555, 38112, 67828, 358274, 372743, 16186423
Offset: 1

Views

Author

N. J. A. Sloane, Aug 16 2004

Keywords

Crossrefs

Cf. A096464, A053300.
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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