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A003881 Decimal expansion of Pi/4. 26
7, 8, 5, 3, 9, 8, 1, 6, 3, 3, 9, 7, 4, 4, 8, 3, 0, 9, 6, 1, 5, 6, 6, 0, 8, 4, 5, 8, 1, 9, 8, 7, 5, 7, 2, 1, 0, 4, 9, 2, 9, 2, 3, 4, 9, 8, 4, 3, 7, 7, 6, 4, 5, 5, 2, 4, 3, 7, 3, 6, 1, 4, 8, 0, 7, 6, 9, 5, 4, 1, 0, 1, 5, 7, 1, 5, 5, 2, 2, 4, 9, 6, 5, 7, 0, 0, 8, 7, 0, 6, 3, 3, 5, 5, 2, 9, 2, 6, 6, 9, 9, 5, 5, 3, 7 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Equals integral_{x=0..infinity} sin(2x)/(2x) dx.

Also equals integral_{x=0..Pi/2} sin(x)^2 dx, or integral_{x=0..Pi/2} cos(x)^2 dx. - Jean-François Alcover, Mar 26 2013

Equals sum(sin(x)*cos(x)/x, x=0..infinity)-1/2. - Bruno Berselli, May 13 2013

Also ratio of area of a circle to the circumscribed square. More generally, ratio of area of an ellipse to the circumscribed rectangle. Also ratio of volume of a cylinder to the circumscribed cube. - Omar E. Pol, Sep 25 2013

Also surface area of a quarter of sphere of diameter 1. - Omar E. Pol, Oct 03 2013

Equals Sum_{n>=0} (-1)^n/(2*n+1). - Geoffrey Critzer, Nov 03 2013

Least positive solution to sin(x) = cos(x). - Franklin T. Adams-Watters, Jun 17 2014

REFERENCES

J. Arndt, Ch. Haenel, "Pi. Algorithmen, Computer, Arithmetik", Springer 2000, p. 150.

D. Hofstadter, "Goedel, Escher, Bach", p. 408.

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

R. K. Hoeflin, Titan Test

Literate Programs, Pi with Machin's formula (Haskell)

S. Ramanujan, Question 353, J. Ind. Math. Soc.

Eric Weisstein's World of Mathematics, Prime Products

Index entries for sequences from "Goedel, Escher, Bach"

FORMULA

Pi/4 = n*A001586(n-1)/A001586(n) as n-->infinity, (conjecture). - Mats Granvik, Feb 23 2011

Pi/4 = integral_{x=0..1} dx/(1+x^2). - Gary W. Adamson, Jun 22 2003

Pi/4 = (-digamma(1/4)+digamma(3/4))/4. [Jean-François Alcover, May 31 2013]

EXAMPLE

0.785398163397448309615660845819875721049292349843776455243736148...

MAPLE

evalf(Pi/4) ;

MATHEMATICA

RealDigits[N[Pi/4, 6! ]] [Vladimir Joseph Stephan Orlovsky, Dec 02 2009]

PROG

(Haskell) see link: Literate Programs

import Data.Char (digitToInt)

a003881_list len = map digitToInt $ show $ machin `div` (10 ^ 10) where

   machin = 4 * arccot 5 unity - arccot 239 unity

   unity = 10 ^ (len + 10)

   arccot x unity = arccot' x unity 0 (unity `div` x) 1 1 where

     arccot' x unity summa xpow n sign

    | term == 0 = summa

    | otherwise = arccot'

      x unity (summa + sign * term) (xpow `div` x ^ 2) (n + 2) (- sign)

    where term = xpow `div` n

-- Reinhard Zumkeller, Nov 20 2012

(Sage) # Leibniz/Cohen/Villegas/Zagier/Arndt/Haenel

def FastLeibniz(n):

    b = 2^(2*n-1); c = b; s = 0

    for k in range(n-1, -1, -1):

        t = 2*k+1

        s = s + c/t if is_even(k) else s - c/t

        b *= (t*(k+1))/(2*(n-k)*(n+k))

        c += b

    return s/c

A003881 = RealField(3333)(FastLeibniz(1330))

print A003881  # Peter Luschny, Nov 20 2012

(PARI) Pi/4 \\ Charles R Greathouse IV, Jul 07 2014

CROSSREFS

Cf. A000796.

Sequence in context: A216542 A216544 A216546 * A225404 A076419 A217516

Adjacent sequences:  A003878 A003879 A003880 * A003882 A003883 A003884

KEYWORD

nonn,cons,easy

AUTHOR

N. J. A. Sloane, Simon Plouffe

EXTENSIONS

a(98) and a(99) corrected. - Reinhard Zumkeller, Nov 20 2012

STATUS

approved

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Last modified August 30 12:08 EDT 2014. Contains 246221 sequences.