A088538 -id:A088538 - cp's OEIS frontend

A278145 Denominator of partial sums of the m=1 member of an m-family of series considered by Hardy with value 4/Pi (see A088538).

1, 8, 64, 1024, 16384, 131072, 1048576, 33554432, 1073741824, 8589934592, 68719476736, 1099511627776, 17592186044416, 140737488355328, 1125899906842624, 72057594037927936, 4611686018427387904, 36893488147419103232, 295147905179352825856, 4722366482869645213696

Offset: 0

Wolfdieter Lang, Nov 14 2016

The numerators seems to coincide with A161736(n+2).
Hardy considered the m-family of series H(m) = 1/m + (1/(m+1))*(1/2)^2 + (1/(m+2))*(1*3/(2*4))^2 + ... = Sum_{k>=0}(1/(m+k))*(risefac(1/2,k)/k!)^2, where risefac(x,m) = Product_{j=0..m-1} (x+j), and risefac(x,0) = 1. See the Hardy reference, p. 106, eq. (7.5.1) (with n=m).
The value of these series H(m) = (Gamma(m) / Gamma(m+1/2))^2 * Sum_{k = 0..m-1} (risefac(1/2,k)/k!)^2.
The present partial sums are for H(1) with value 1/Gamma(3/2)^2 = 4/Pi (A088538).

G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, p. 106, eq. (7.5.1), and references on p. 112 for Darling (1), p. 232, and Watson (5), p. 235.

Cf. A000165, A001147, A088538, A161736.

Table[Denominator@ Sum[(1/(k + 1)) (Pochhammer[1/2, k]/k!)^2, {k, 0, n}], {n, 0, 19}] (* or *)
Table[Denominator@ Sum[(1/(k + 1)) (Binomial[-1/2, k])^2, {k, 0, n}], {n, 0, 19}] (* or *)
Table[Denominator@ Sum[(1/(k + 1)) ((2 k - 1)!!/(2 k)!!)^2, {k, 0, n}], {n, 0, 19}] (* Michael De Vlieger, Nov 15 2016 *)

a(n)= denominator(r(n)) with the rationals r(n) = Sum_{k=0..n}(1/(k+1))*(risefac(1/2,k)/k!)^2 = Sum_{k=0..n} (1/(k+1))*(binomial(-1/2,k))^2 = Sum_{k=0..n}(1/(k+1))*((2*k-1)!!/(2*k)!!)^2 , with the rising factorial risefac(x,k) defined above. The double factorials are given in A001147 and A000165 with (-1)!! := 1.

A049541 Decimal expansion of 1/Pi.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Dec 11 1999

The ratio of the volume of a regular octahedron to the volume of the circumscribed sphere (which has circumradius a*sqrt(2)/2 = a*A010503, where a is the octahedron's edge length; see MathWorld link). For similar ratios for other Platonic solids, see A165922, A165952, A165953 and A165954. - Rick L. Shepherd, Oct 01 2009

Corresponds to a gauge point marked "M" on slide rule calculating devices in the 20th century. The Pickworth reference notes its use in calculating the area of the curved surface of a cylinder. - Peter Munn, Aug 14 2020

			0.3183098861837906715377675267450287240689192914809128974953...

J.-P. Delahaye, Pi - die Story (German translation), Birkhäuser, 1999 Baasel, p. 245. French original: Le fascinant nombre Pi, Pour la Science, Paris, 1997.
C. N. Pickworth, The Slide Rule, 24th Ed., Pitman, London, 1945, p. 53, Gauge Points.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, p. 27.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Mohammad K. Azarian, An Expression for Pi, Problem #870, College Mathematics Journal, Vol. 39, No. 1, January 2008, p. 66. Solution appeared in Vol. 40, No. 1, January 2009, pp. 62-64.
J. Bohr, Ramanujan's Method of Approximating Pi.
J. Borwein, Ramanujan's Sum.
Heng Huat Chan, Shaun Cooper, and Wen-Chin Liaw, The Rogers-Ramanujan continued fraction and a quintic iteration for 1/Pi, Proc. Amer. Math. Soc. 135 (2007), 3417-3424.
D. V. Chudnovsky and G. V. Chudnovsky, The computation of classical constants, Proc. Nati. Acad. Sci. USA, Vol. 86, pp. 8178-8182, November 1989.
J. Guillera, A New Method to Obtain Series for 1/Pi and 1/Pi^2, Experimental Mathematics, Volume 15, Issue 1, 2006.
R. Matsumoto, Ramanujan Type Series. [Broken link]
A. S. Nimbran, Deriving Forsyth-Glaisher type series for 1/π and Catalan's constant by an elementary method, The Mathematics Student, Indian Math. Soc., Vol. 84, Nos. 1-2, Jan.-June (2015), 69-86. [Broken link]
Eric W. Weisstein, Octahedron.
Index entries for transcendental numbers.

Cf. A000796, A165922, A165952, A165953, A165954, A063723, A010503. - Rick L. Shepherd, Oct 01 2009

Cf. A088538 (4/Pi).

MATLAB
```
1/pi % Altug Alkan, Apr 10 2016
```

R:= RealField(100); 1/Pi(R); // G. C. Greubel, Aug 21 2018

Digits:=100: evalf(1/Pi); # Wesley Ivan Hurt, Aug 29 2016

RealDigits[N[1/Pi, 10, 100]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jun 18 2009 *)

1/Pi \\ Charles R Greathouse IV, Jun 16 2011

Equals (1/(12-16*A002162))*Sum_{n>=0} A002894(n)*H(n)/(A001025(n) * A016754(n-1)), where H(n) denotes the n-th harmonic number. - John M. Campbell, Aug 28 2016
1/Pi = Sum_{m>=0} binomial(2*m, m)^3 * (42*m+5)/(2^(12*m+4)), Ramanujan, from the J.-P. Delahaye reference. - Wolfdieter Lang, Sep 18 2018; corrected by Bernard Schott, Mar 26 2020
1/Pi = 12*Sum_{n >= 0} (-1)^n*((6*n)!/(n!^3*(3*n)!))*(13591409 + 545140134*n)/640320^(3*n + 3/2) [Chudnovsky]. - Sanjar Abrarov, Mar 31 2020
1/Pi = (sqrt(8)/9801) * Sum_{n >= 0} ((4*n)!/((n!)^4)) * (26390*n + 1103)/(396^(4*n)) [Ramanujan, 1914]. - Bernard Schott, Mar 26 2020
Equal Sum_{k>=2} tan(Pi/2^k)/2^k. - Amiram Eldar, Aug 05 2020
Floor((3/8)*Sum_{n>=1} sigma[3](n)*n/exp(Pi*n/(10^((1/5)*k+(1/5))))) mod 10, will give the k-th digit of 1/Pi. - Simon Plouffe, Dec 19 2023

A033996 8 times triangular numbers: a(n) = 4n(n+1).

Original entry on oeis.org

0, 8, 24, 48, 80, 120, 168, 224, 288, 360, 440, 528, 624, 728, 840, 960, 1088, 1224, 1368, 1520, 1680, 1848, 2024, 2208, 2400, 2600, 2808, 3024, 3248, 3480, 3720, 3968, 4224, 4488, 4760, 5040, 5328, 5624, 5928, 6240, 6560, 6888, 7224, 7568, 7920, 8280

Offset: 0

N. J. A. Sloane, Dec 11 1999

Write 0, 1, 2, ... in a clockwise spiral; sequence gives numbers on one of 4 diagonals.
Also, least m > n such that T(m)*T(n) is a square and more precisely that of A055112(n). {T(n) = A000217(n)}. - Lekraj Beedassy, May 14 2004
Also sequence found by reading the line from 0, in the direction 0, 8, ... and the same line from 0, in the direction 0, 24, ..., in the square spiral whose vertices are the generalized decagonal numbers A074377. Axis perpendicular to A195146 in the same spiral. - Omar E. Pol, Sep 18 2011
Number of diagonals with length sqrt(5) in an (n+1) X (n+1) square grid. Every 1 X 2 rectangle has two such diagonals. - Wesley Ivan Hurt, Mar 25 2015
Imagine a board made of squares (like a chessboard), one of whose squares is completely surrounded by square-shaped layers made of adjacent squares. a(n) is the total number of squares in the first to n-th layer. a(1) = 8 because there are 8 neighbors to the unit square; adding them gives a 3 X 3 square. a(2) = 24 = 8 + 16 because we need 16 more squares in the next layer to get a 5 X 5 square: a(n) = (2*n+1)^2 - 1 counting the (2n+1) X (2n+1) square minus the central square. - R. J. Cano, Sep 26 2015
The three platonic solids (the simplex, hypercube, and cross-polytope) with unit side length in n dimensions all have rational volume if and only if n appears in this sequence, after 0. - Brian T Kuhns, Feb 26 2016
The number of active (ON, black) cells in the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 645", based on the 5-celled von Neumann neighborhood. - Robert Price, May 19 2016
The square root of a(n), n>0, has continued fraction [2n; {1,4n}] with whole number part 2n and periodic part {1,4n}. - Ron Knott, May 11 2017
Numbers k such that k+1 is a square and k is a multiple of 4. - Bruno Berselli, Sep 28 2017
a(n) is the number of vertices of the octagonal network O(n,n); O(m,n) is defined by Fig. 1 of the Siddiqui et al. reference. - Emeric Deutsch, May 13 2018
a(n) is the number of vertices in conjoined n X n octagons which are arranged into a square array, a.k.a. truncated square tiling. - Donghwi Park, Dec 20 2020
a(n-2) is the number of ways to place 3 adjacent marks in a diagonal, horizontal, or vertical row on an n X n tic-tac-toe grid. - Matej Veselovac, May 28 2021

			Spiral with 0, 8, 24, 48, ... along lower right diagonal:
.
  36--37--38--39--40--41--42
   |                       |
  35  16--17--18--19--20  43
   |   |               |   |
  34  15   4---5---6  21  44
   |   |   |       |   |   |
  33  14   3   0   7  22  45
   |   |   |   | \ |   |   |
  32  13   2---1   8  23  46
   |   |           | \ |   |
  31  12--11--10---9  24  47
   |                   | \ |
  30--29--28--27--26--25  48
                            \
[Reformatted by _Jon E. Schoenfield_, Dec 25 2016]

Stuart M. Ellerstein, J. Recreational Math. 29 (3) 188, 1998.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd ed., 1994, p. 99.
Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
M. K. Siddiqui, M. Naeem, N. A. Rahman, and M. Imran, Computing topological indices of certain networks, J. of Optoelectronics and Advanced Materials, 18, No. 9-10, 2016, 884-892.
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Leo Tavares, Illustration: Centroid Diamonds.
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton.
Eric Weisstein's World of Mathematics, Hamiltonian Path.
Eric Weisstein's World of Mathematics, Knight Graph.
Stephen Wolfram, A New Kind of Science
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata.
Index to Elementary Cellular Automata.

Cf. A000217, A016754, A002378, A024966, A027468, A028895, A028896, A045943, A046092, A049598, A088538, A124080, A008590 (first differences), A130809 (partial sums).
Sequences from spirals: A001107, A002939, A002943, A007742, A033951, A033952, A033953, A033954, A033988, A033989, A033990, A033991, A033996. - Omar E. Pol, Dec 12 2008
Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.
Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.

[ 4*n*(n+1) : n in [0..50] ]; // Wesley Ivan Hurt, Jun 09 2014

seq(8*binomial(n+1, 2), n=0..46); # Zerinvary Lajos, Nov 24 2006
[seq((2*n+1)^2-1, n=0..46)];

Table[(2n - 1)^2 - 1, {n, 50}] (* Alonso del Arte, Mar 31 2013 *)

nsqm1(n) = { forstep(x=1,n,2, y = x*x-1; print1(y, ", ") ) }

a(n) = 4*n^2 + 4*n = (2*n+1)^2 - 1.
G.f.: 8*x/(1-x)^3.
a(n) = A016754(n) - 1 = 2*A046092(n) = 4*A002378(n). - Lekraj Beedassy, May 25 2004
a(n) = A049598(n) - A046092(n); a(n) = A124080(n) - A002378(n). - Zerinvary Lajos, Mar 06 2007
a(n) = 8*A000217(n). - Omar E. Pol, Dec 12 2008
a(n) = A005843(n) * A163300(n). - Juri-Stepan Gerasimov, Jul 26 2009
a(n) = a(n-1) + 8*n (with a(0)=0). - Vincenzo Librandi, Nov 17 2010
For n > 0, a(n) = A058031(n+1) - A062938(n-1). - Charlie Marion, Apr 11 2013
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Wesley Ivan Hurt, Mar 25 2015
a(n) = A000578(n+1) - A152618(n). - Bui Quang Tuan, Apr 01 2015
a(n) - a(n-1) = A008590(n), n > 0. - Altug Alkan, Sep 26 2015
From Ilya Gutkovskiy, May 19 2016: (Start)
E.g.f.: 4*x*(2 + x)*exp(x).
Sum_{n>=1} 1/a(n) = 1/4. (End)
Product_{n>=1} a(n)/A016754(n) = Pi/4. - Daniel Suteu, Dec 25 2016
a(n) = A056220(n) + A056220(n+1). - Bruce J. Nicholson, May 29 2017
sqrt(a(n)+1) - sqrt(a(n)) = (sqrt(n+1) - sqrt(n))^2. - Seiichi Manyama, Dec 23 2018
a(n)*a(n+k) + 4*k^2 = m^2 where m = (a(n) + a(n+k))/2 - 2*k^2; for k=1, m = 4*n^2 + 8*n + 2 = A060626(n). - Ezhilarasu Velayutham, May 22 2019
Sum_{n>=1} (-1)^n/a(n) = 1/4 - log(2)/2. - Vaclav Kotesovec, Dec 21 2020
From Amiram Eldar, Feb 21 2023: (Start)
Product_{n>=1} (1 - 1/a(n)) = -(4/Pi)*cos(Pi/sqrt(2)).
Product_{n>=1} (1 + 1/a(n)) = 4/Pi (A088538). (End)

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

Jason Earls, Mar 28 2001

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

			2/Pi = 0.6366197723675813430755350534900574481378385829618257949906...

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.

Harry J. Smith, Table of n, a(n) for n = 0..20000
Iaroslav V. Blagouchine and Eric Moreau, On a finite sum of cosecants appearing in various problems, Journal of Mathematical Analysis and Applications, vol. 539, no. 1, pt. 2, pp. 1-36, 2024. See p. 18.
K. S. Brown, MathPages: The Algebra of an Infinite Grid of Resistors
G. Buffon, Essai d'arithmétique morale, Supplément à l'Histoire Naturelle, Vol. 4, 1777.
Encyclopedia of Mathematics, Arcsine distribution
Boris Gourevitch, L'univers de Pi
Mark Kac, On the average number of real roots of a random algebraic equation, Bull. Amer. Math. Soc. 49:4 (1943), pp. 314-320.
MacTutor History of Mathematics archive, Georges-Louis Leclerc, Comte de Buffon.
Veikko Nevanlinna, On constants connected with the prime number theorem for arithmetic progressions, Annales Academiae Scientiarum Fennicae Ser. A. I., No. 539 (1973).
Da-Wei Niu, Jian Cao, and Feng Qi, Generalizations of Jordan's inequality and concerned relations, U.P.B. Sci. Bull., Series A, Vol. 72, Iss. 3, 2010
Herbert Solomon, Geometric Probability, SIAM, 1978, p. 152. [See average chord length comment]
Eric Weisstein's World of Mathematics, Buffon's needle problem.
Eric Weisstein's World of Mathematics, Magic Geometric Constants.
Eric Weisstein's World of Mathematics, Prime Products.
Eric Weisstein's World of Mathematics, Geometric Centroid.
Eric Weisstein's World of Mathematics, Jordan's Inequality.
Wikipedia, Buffon's needle problem.
Wikipedia, Centroid.
Wikipedia, Jordan's inequality.
Index entries for transcendental numbers.

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

Cf. A076668 (sqrt(2/Pi)).

R:= RealField(100); 2/Pi(R); // G. C. Greubel, Mar 09 2018

Digits:=100: evalf(2/Pi); # Wesley Ivan Hurt, Aug 02 2014

Mathematica
```
RealDigits[ N[ 2/Pi, 111]][[1]]
```

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

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)

A086201 Decimal expansion of 1/(2*Pi).

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Jul 12 2003

If a single hump of cycloid, with arc length 8*radius (generating circle), is inside a rectangle with width=2*radius and length=2*Pi*radius, then the radius must be 1/(2*Pi) (this sequence) to have (2/Pi), A060294, as semi arc of cycloid (arc = 4/Pi = A088538) and the rectangle... length = 1, width = 1/Pi. I suppose that in 3D geometry, gliding along a cycloid, in all directions around, from a point A at the height of 1/Pi, gives Pi*point B. - Eric Desbiaux, Dec 21 2008
Radius of circle having circumference 1. - Clark Kimberling, Jan 06 2014
The number of primitive Pythagorean triangles with hypotenuse less than N is approximately N/(2*Pi), found by Lehmer, cf. Knott link. - Frank Ellermann, Mar 27 2020

			0.15915494309189533576888376337251...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.4, p. 493.

Ron Knott, 9.6 Pythagorean Triples and Pi, 2019.
Eric Weisstein's World of Mathematics, Plouffe's Constants.
Eric Weisstein's World of Mathematics, Pythagorean Triple.
Index entries for transcendental numbers.

Cf. A000796 (Pi), A019692 (2*Pi).

RealDigits[N[1/(2 Pi), 100]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jun 18 2009 *)

PARI
```
1/(2*Pi) \\ Michel Marcus, Mar 28 2020
```

Link corrected by Fred Daniel Kline, Jul 29 2015

A070750 0 if n-th prime is even, 1 if n-th prime is == 1 (mod 4), and -1 if n-th prime is == 3 (mod 4).

Original entry on oeis.org

0, -1, 1, -1, -1, 1, 1, -1, -1, 1, -1, 1, 1, -1, -1, 1, -1, 1, -1, -1, 1, -1, -1, 1, 1, 1, -1, -1, 1, 1, -1, -1, 1, -1, 1, -1, 1, -1, -1, 1, -1, 1, -1, 1, 1, -1, -1, -1, -1, 1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 1, -1, 1, -1, -1, 1, 1, -1, 1, -1, 1, 1, -1

Offset: 1

Reinhard Zumkeller, May 04 2002

Also, sin(prime(n)*Pi/2), where prime(n) = A000040(n), Pi=3.1415... (original definition).
Also imaginary part of primes mapped as defined in A076340, A076341: a(n) = A076341(A000040(n)), real part = A076342.
Legendre symbol (-1/prime(n)) for n > 1. - T. D. Noe, Nov 05 2003
For n > 1, let p = prime(n) and m = (p-1)/2. Then c(m) - a(n) == 0 (mod p), where c(m) = (2*m)!/(m!)^2 = A000984(m) is the central binomial coefficient. [Proof: By definition, c(m)*(m!)^2 - (p-1)! = 0 and therefore c(m)*(m!)^2*(-1)^(m+1) - (p-1)!*(-1)^(m+1) = 0. Now apply Wilson's theorem, (p-1)! == 1 (mod p), and its corollary, (m!)^2 == (-1)^(m+1) (mod p), and finally use the formula by T. D. Noe listed below to replace (-1)^m with a(n).] Similarly, C_m - 2*a(n) == 0 (mod p), with C_m = A000108(m) being the m-th Catalan number. [Proof: By definition, C_m*(p+1)*(m!)^2 - 2*(p-1)! = 0. The result follows proceeding as in the first proof.] - Stanislav Sykora, Aug 11 2014

			p = 4*k+1 (see A002144): a(p) = sin((4*k+1)*Pi/2) = sin(2*k*Pi + Pi/2) = sin(Pi/2) = 1.
p = 4*k+3 (see A002145): a(p) = sin((4*k+3)*Pi/2) = sin(2*k*Pi + 3*Pi/2) = sin(3*Pi/2) = -1.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Legendre Symbol.
Wikipedia, Wilson's theorem.

Cf. A000040, A039702, A070748, A070749, A002144, A002145, A000108, A000984, A134323, A257834, A076340, A076341.

Cf. A060294, A088538.

a070750 = (2 -) . (`mod` 4) . a000040  -- Reinhard Zumkeller, Feb 28 2012

a[n_] := JacobiSymbol[-1, Prime[n]]; a[1] = 0; Table[a[n], {n, 1, 72}] (* Jean-François Alcover, Oct 05 2012, after T. D. Noe *)
Table[Which[EvenQ[p],0,Mod[p,4]==1,1,True,-1],{p,Prime[Range[80]]}] (* Harvey P. Dale, Mar 16 2020 *)

apply(n->2-n%4,primes(100)) \\ Charles R Greathouse IV, Aug 21 2011

a(n) = 2 - prime(n) mod 4 = 2 - A039702(n).
a(n) = (-1)^((prime(n)-1)/2) for n > 1. - T. D. Noe, Nov 05 2003
From Amiram Eldar, Dec 24 2022: (Start)
Product_{n>=1} (1 - a(n)/prime(n)) = 4/Pi (A088538).
Product_{n>=1} (1 + a(n)/prime(n)) = 2/Pi (A060294). (End)

Wording of definition changed by N. J. A. Sloane, Jun 21 2015

A132696 Decimal expansion of 6/Pi.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 26 2007, Nov 02 2007

6/Pi = Volume of the cuboid (If L1>L2>L3) / Volume of the inscribed ellipsoid.
6/Pi = Volume of the cuboid (If L1>(L2=L3)) / Volume of the inscribed spheroid.
6/Pi = Volume of the regular hexahedron (or cube) / Volume of the inscribed Sphere.
6/Pi = 1 / Arc of 30 degrees.
6/Pi = Volume of the cuboid (If L1<(L2=L3)) / Volume of the inscribed spheroid.
6/Pi = Surface area of the regular hexahedron (or cube) / surface area of the inscribed sphere.

			1.90985931710274402922660516047... .

Index entries for transcendental numbers

Cf. A049541, A060294, A089491, A088538, A086201, A059956.

RealDigits[N[6/Pi, 200]] (* Erich Friedman, Mar 22 2008 *)

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

Equals Product_{k>=1} (2k+1)^3 / ( (2k)^2*(2k+3) ). - Federico Provvedi, Nov 09 2024

More terms from Erich Friedman, Mar 22 2008

A000888 a(n) = (2n)!^2 / ((n+1)!n!^3).

Original entry on oeis.org

1, 2, 12, 100, 980, 10584, 121968, 1472328, 18404100, 236390440, 3103161776, 41469525552, 562496897872, 7726605740000, 107289439704000, 1503840313184400, 21252802073091300, 302539888334593800, 4334635827016110000, 62464383654579522000, 904841214653480504400

Offset: 0

N. J. A. Sloane

a(n) is the number of walks of 2n unit steps North, East, South, or West, starting at the origin, bounded above by y=x, below by y=-x and terminating on the ray y = x >= 0. Example: a(1) counts EN, EW; a(2) counts ESNN, ESNW, ENSN, ENSW, ENEN, ENEW, EENN, EENW, EEWN, EEWW, EWEN, EWEW. - David Callan, Oct 11 2005
Bijective proof: given such an NESW walk, construct a pair (P_1, P_2) of lattice paths of upsteps U=(1,1) and downsteps D=(1,-1) as follows. To get P_1, replace each E and S with U and each W and N with D. To get P_2, replace each N and E with U and each S and W with D. For example, EENSNW -> (UUDUDD, UUUDUD). This mapping is 1-to-1 and its range is the Cartesian product of the set of Dyck n-paths and the set of nonnegative paths of length 2n. The Dyck paths are counted by the Catalan number C_n (A000108) and the nonnegative paths are counted (see for example the Callan link) by the central binomial coefficient binomial(2n,n) (A000984). So this is a bijection from these NESW walks to a set of size C_n*binomial(2n,n) = a(n). - David Callan, Sep 18 2007
If A is a random matrix in USp(4) (4 X 4 complex matrices that are unitary and symplectic), then a(n) = E[(tr(A^3))^{2n}]. - Andrew V. Sutherland, Apr 01 2008
Number of walks within N^2 (the first quadrant of Z^2) starting at (0,0), ending on the vertical axis and consisting of 2 n steps taken from {(-1,-1), (-1,1), (1,-1), (1,1)}. - Manuel Kauers, Nov 18 2008
a(n) is equal to the n-th moment of the following positive function defined on x in (0,16), in Maple notation: (EllipticK(sqrt(1-x/16)) - EllipticE(sqrt(1-x/16)))/(Pi^2*sqrt(x)). This is the solution of the Hausdorff moment problem and thus it is unique. - Karol A. Penson, Feb 11 2011
The partial sums of a(n)/A013709(n) absolutely converge to 1/Pi. - Ralf Steiner, Jan 21 2016

			G.f.: 1 + 2*x + 12*x^2 + 100*x^3 + 980*x^4 + 10584*x^5 + 121968*x^6 + ...

E. R. Hansen, A Table of Series and Products, Prentice-Hall, Englewood Cliffs, NJ, 1975, p. 93.
T. M. MacRobert, Functions of a Complex Variable, 4th ed., Macmillan & Co., London, 1958, p. 177.

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Marco S. Bianchi, Protected and uniformly transcendental, arXiv:2306.06239 [hep-th], 2023.
M. Bousquet-Mélou and M. Mishna, Walks with small steps in the quarter plane, arXiv:0810.4387 [math.CO], 2008-2009.
David Callan, Bijections for the identity 4^n = ... .
Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices, arXiv:0803.4462 [math.NT], 2008-2010.
Helmut Prodinger, Two New Identities Involving the Catalan Numbers: A classical approach, arXiv:1911.07604 [math.CO], 2019.
Ralf Steiner, Beispiele zur modifizierten Wallis-Lambert-Reihe, 2016.

Cf. A000108, A002894, A088538, A089835, A125558.

[(Factorial(2*n))^2/(Factorial(n))^4/(n+1): n in [0..20]]; // Vincenzo Librandi, Aug 15 2011

[seq(binomial(2*n,n)^2/(n+1),n=0..17)]; # Zerinvary Lajos, May 27 2006

f[n_] := Binomial[2 n, n]^2/(n + 1); Array[f, 18, 0]  (* Robert G. Wilson v *)
a[ n_] := SeriesCoefficient[ (1/8) (EllipticE[ 16 x] - (1 - 16 x) EllipticK[ 16 x]) / (Pi/2), {x, 0, n + 1}]; (* Michael Somos, Jan 23 2012 *)

{a(n) = if( n<0, 0, (2*n)!^2 / n!^4 / (n+1))}; /* Michael Somos, Sep 11 2005 */

G.f.: 1/4*((16*x-1)*EllipticK(4*x^(1/2)) + EllipticE(4*x^(1/2)))/x/Pi. - Vladeta Jovovic, Oct 12 2003
Given G.f. A(x), y = x*A(x) satisfies y = y'' * (1 - 16*x) * x/4. - Michael Somos, Sep 11 2005
a(n) = binomial(2*n,n)^2/(n+1). - Zerinvary Lajos, May 27 2006
G.f.: 2F1(1/2,1/2;2;16*x). - Paul Barry, Sep 03 2008
a(n) = 2*A125558(n) (n >= 1). - Olivier Gérard, Feb 16 2011
A002894(n) = (n+1) * a(n). A001246(n) = a(n) / (n+1). A089835(n) = n! * a(n). - Michael Somos, May 12 2012
G.f.: 1 + 4*x/(G(0)-4*x) where G(k) = 4*x*(2*k+1)^2 + (k+1)*(k+2) - 4*x*(k+1)*(k+2)*(2*k+3)^2/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Jul 30 2012
D-finite with recurrence: (n+1)*(n+2)*a(n+1) = 4*(2*n+1)^2*a(n). - Vaclav Kotesovec, Sep 11 2012
a(n) = C(n)*binomial(2*n,n) = Sum_{k=0..2*n} binomial(2*n,k)*C(k)*C(2*n-k) where C(k) are Catalan numbers (A000108), see Prodinger. - Michel Marcus, Nov 19 2019
Sum_{n>=0} a(n)/16^n = 4/Pi (A088538). - Amiram Eldar, May 06 2023

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

cp's OEIS Frontend

A278145 Denominator of partial sums of the m=1 member of an m-family of series considered by Hardy with value 4/Pi (see A088538).

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A049541 Decimal expansion of 1/Pi.

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A033996 8 times triangular numbers: a(n) = 4*n*(n+1).

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

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A086201 Decimal expansion of 1/(2*Pi).

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A070750 0 if n-th prime is even, 1 if n-th prime is == 1 (mod 4), and -1 if n-th prime is == 3 (mod 4).

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A033996 8 times triangular numbers: a(n) = 4n(n+1).

A000888 a(n) = (2n)!^2 / ((n+1)!n!^3).