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A088538 Decimal expansion of 4/Pi. 32
1, 2, 7, 3, 2, 3, 9, 5, 4, 4, 7, 3, 5, 1, 6, 2, 6, 8, 6, 1, 5, 1, 0, 7, 0, 1, 0, 6, 9, 8, 0, 1, 1, 4, 8, 9, 6, 2, 7, 5, 6, 7, 7, 1, 6, 5, 9, 2, 3, 6, 5, 1, 5, 8, 9, 9, 8, 1, 3, 3, 8, 7, 5, 2, 4, 7, 1, 1, 7, 4, 3, 8, 1, 0, 7, 3, 8, 1, 2, 2, 8, 0, 7, 2, 0, 9, 1, 0, 4, 2, 2, 1, 3, 0, 0, 2, 4, 6, 8, 7, 6, 4, 8, 5, 8 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Average length of chord formed from two randomly chosen points on the circumference of a unit circle (see Weisstein/MathWorld link). - Rick L. Shepherd, Jun 19 2006

Suppose u(0) = 1 + i where i^2 = -1 and u(n+1) = (1/2)*(u(n) + |u(n)|). Conjecture: limit(Real(u(n)), n = +infinity) = 4/Pi. - Yalcin Aktar, Jul 18 2007

Ratio of the arc length of the cycloid for one period to the circumference of the corresponding circle rolling on a line. Thus, for any integral number n of revolutions of a circle of radius r, a point on the circle travels 4/Pi*2Pi*r*n = 8rn (while the center of the circle moves only 2Pi*rn). This ratio varies for partial revolutions and depends upon the initial position of the point with points nearest the line moving the slowest (see Dudeney, who explains how the tops of bicycle wheels move faster than the parts nearest the ground). - Rick L. Shepherd, May 05 2014

Average distance travelled in two steps of length 1 for a random walk in the plane starting at the origin. - Jean-Fran├žois Alcover, Aug 04 2014

REFERENCES

H. E. Dudeney, 536 Puzzles & Curious Problems, Charles Scribner's Sons, New York, 1967, pp. 99, 300-301, #294.

S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, p. 86

L. B. W. Jolley, Summation of Series, Dover (1961).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

J.-P. Allouche, On a formula of T. Rivoal, arXiv:1307.3906 [math.NT], 2013.

J. M. Borwein, A. Straub, J. Wan, and W. Zudilin, Densities of short uniform random walks, arXiv:1103.2995 [math.CA], 2011.

R. J. Mathar, Chebyshev Series Expansion of Inverse Polynomials, arXiv:0403344 [math.CA]

Eric Weisstein's World of Mathematics, Circle Line Picking.

Eric Weisstein's World of Mathematics, Cycloid.

FORMULA

4/Pi = prod(1-(-1)^((p-1)/2)/p) where p runs through the odd primes.

arcsin x = (4/Pi) sum_{n = 1, 3, 5, 7, ...} T_n(x)/n^2 (Chebyshev series of arcsin; App C of math.CA/0403344). - R. J. Mathar, Jun 26 2006

Equals 1 + sum_{n >= 1} ((2n-3)!!/(2n)!!)^2. [Jolley eq 274]. - R. J. Mathar, Nov 03 2011

EXAMPLE

4/Pi = 1.2732395.... = 1/0.78539816...

MATHEMATICA

RealDigits[N[4/Pi, 6!]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jun 18 2009 *)

PROG

(PARI) 4/Pi \\ Charles R Greathouse IV, Jun 21 2013

CROSSREFS

Cf. A079097 for terms of a generalized continued fraction for 4/Pi.

Sequence in context: A124910 A090388 A021370 * A210516 A226626 A011049

Adjacent sequences:  A088535 A088536 A088537 * A088539 A088540 A088541

KEYWORD

cons,nonn

AUTHOR

Benoit Cloitre, Nov 16 2003

STATUS

approved

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Last modified October 1 23:55 EDT 2014. Contains 247527 sequences.