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A003881 Decimal expansion of Pi/4. 49
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_{x=0..infinity} sin(x)*cos(x)/x) - 1/2. - Bruno Berselli, May 13 2013

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

Also the surface area of a quarter-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

Equals Integral_{x=0..1} Product_{k>=1} (1-x^(8*k))^3 dx [Cf. A258414]. - Vaclav Kotesovec, May 30 2015

Dirichlet L-series of the non-principal character modulo 4 (A101455) at 1. See e.g. Table 22 of arXiv:1008.2547. - R. J. Mathar, May 27 2016

This constant is also equal to the infinite sum of the arctangent functions with nested radicals consisting of square roots of two. Specifically, one of the Viete-like formulas for Pi is given by Pi/4 = Sum_{k = 2..infinity} arctan(sqrt(2 - a_{k - 1})/a_k), where the nested radicals are defined by recurrence relations a_k = sqrt(2 + a_{k - 1}) and a_1 = sqrt(2) (see the article [Abrarov and Quine]). - Sanjar Abrarov, Jan 09 2017

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

S. M. Abrarov and B. M. Quine, A Viète-like formula for pi based on infinite sum of the arctangent functions with nested radicals, figshare, 4509014, (2017).

J. M. Borwein, P. B. Borwein, K. Dilcher, Pi, Euler numbers and asymptotic expansions, Amer. Math. Monthly, 96 (1989), 681-687.

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

Wikipedia, Leibniz formula for Pi

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]

Pi/4 = Product_{k \in A071904} (if k mod 4 == 1 then (k-1)/(k+1)) else (if k mod 4 == 3 then (k+1)/(k-1)). - Dimitris Valianatos, Oct 05 2016

From Peter Bala, Nov 15 2016: (Start)

For N even: 2*(Pi/4 - Sum_{k = 1..N/2} (-1)^(k-1)/(2*k - 1)) ~ (-1)^(N/2)*(1/N - 1/N^3 + 5/N^5 - 61/N^7 + 1385/N^9 - ...), where the sequence of unsigned coefficients [1, 1, 5, 61, 1385, ...] is A000364. See Borwein et al., Theorem 1 (a).

For N odd: 2*(Pi/4 - Sum_{k = 1..(N-1)/2} (-1)^(k-1)/(2*k - 1)) ~ (-1)^((N-1)/2)*(1/N - 1/N^2 + 2/N^4 - 16/N^6 + 272/N^8 - ...), where the sequence of unsigned coefficients [1, 1, 2, 16, 272, ...] is A000182 with an extra initial term of 1.

For N = 0,1,2,... and m = 1,3,5,... there holds Pi/4 = (2*N)! * m^(2*N) * Sum{k >= 0} ( (-1)^(N+k) * 1/Product_{j = -N..N} (2*k + 1 + 2*m*j) ); when N = 0 we get the Madhava-Gregory-Leibniz series for Pi/4.

For examples of asymptotic expansions for the tails of these series representations for Pi/4 see A024235 (case N = 1, m = 1), A278080 (case N = 2, m = 1) and A278195 (case N = 3, m = 1).

For N = 0,1,2,..., Pi/4 = 4^(N-1)*N!/(2*N)! * Sum_{k >= 0} 2^(k+1)*(k + N)!* (k + 2*N)!/(2*k + 2*N + 1)!, follows by applying Euler's series transformation to the above series representation for Pi/4 in the case m = 1. (End)

EXAMPLE

0.785398163397448309615660845819875721049292349843776455243736148...

N = 2, m = 6: Pi/4 = 4!*3^4 Sum_{k >= 0} (-1)^k/((2*k - 11)*(2*k - 5)*(2*k + 1)*(2*k + 7)*(2*k + 13)). - Peter Bala, Nov 15 2016

MAPLE

evalf(Pi/4) ;

MATHEMATICA

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

(* PROGRAM STARTS *)

(* Define the nested radicals a_k by recurrence *)

a[k_] := Nest[Sqrt[2 + #1] & , 0, k]

(* Example of Pi/4 approximation at K = 100 *)

Print["The actual value of Pi/4 is"]

N[Pi/4, 40]

Print["At K = 100 the approximated value of Pi/4 is"]

K := 100;  (* the truncating integer *)

N[Sum[ArcTan[Sqrt[2 - a[k - 1]]/a[k]], {k, 2, K}], 40] (* equation (8) *)

(* Error terms for Pi/4 approximations *)

Print["Error terms for Pi/4"]

k := 1; (* initial value of the index k *)

K := 10; (* initial value of the truncating integer K *)

sqn := {}; (* initiate the sequence *)

AppendTo[sqn, {"Truncating integer K ", " Error term in Pi/4"}];

While[K <= 30,

AppendTo[sqn, {K,

   N[Pi/4 - Sum[ArcTan[Sqrt[2 - a[k - 1]]/a[k]], {k, 2, K}], 1000] //

    N}]; K++]

Print[MatrixForm[sqn]]

(* Sanjar Abrarov, Jan 09 2017 *)

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, A001586, A071904.

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 by Reinhard Zumkeller, Nov 20 2012

STATUS

approved

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Last modified May 25 10:24 EDT 2017. Contains 287026 sequences.