

A019669


Decimal expansion of Pi/2.


97



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
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OFFSET

1,2


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


LINKS



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
The previous result is the particular case n = 1 of the more general identity: Pi/2 = 4^(n1) * 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 = 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(n1)), 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 AbelPlana formula.  Peter Bala, Nov 05 2019
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)^((p1)/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


EXAMPLE

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


MAPLE



MATHEMATICA



PROG

(PARI) default(realprecision, 20080); x=Pi/2; for (n=1, 20000, d=floor(x); x=(xd)*10; write("b019669.txt", n, " ", d)); \\ Harry J. Smith, May 31 2009


CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



