

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.  JeanFrançois Alcover, Aug 04 2014


REFERENCES

H. E. Dudeney, 536 Puzzles & Curious Problems, Charles Scribner's Sons, New York, 1967, pp. 99, 300301, #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)^((p1)/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} ((2n3)!!/(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: A090388 A021370 A248140 * A210516 A226626 A249778
Adjacent sequences: A088535 A088536 A088537 * A088539 A088540 A088541


KEYWORD

cons,nonn


AUTHOR

Benoit Cloitre, Nov 16 2003


STATUS

approved



