%I #199 Oct 05 2024 11:35:58

%S 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,

%T 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,

%U 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

%N Decimal expansion of Pi/4.

%C 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

%C Also the surface area of a quarter-sphere of diameter 1. - _Omar E. Pol_, Oct 03 2013

%C Least positive solution to sin(x) = cos(x). - _Franklin T. Adams-Watters_, Jun 17 2014

%C 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

%C 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..oo} 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

%C Pi/4 is the area enclosed between circumcircle and incircle of a regular polygon of unit side. - _Mohammed Yaseen_, Nov 29 2023

%D Jörg Arndt and Christoph Haenel, Pi: Algorithmen, Computer, Arithmetik, Springer 2000, p. 150.

%D Douglas R. Hofstadter, Gödel, Escher, Bach: An Eternal Golden Braid, Basic Books, p. 408.

%D J. Rivaud, Analyse, Séries, équations différentielles, Mathématiques supérieures et spéciales, Premier cycle universitaire, Vuibert, 1981, Exercice 3, p. 136.

%H Reinhard Zumkeller, <a href="/A003881/b003881.txt">Table of n, a(n) for n = 0..1000</a>

%H Sanjar M. Abrarov and Brendan M. Quine, <a href="https://dx.doi.org/10.6084/m9.figshare.4509014">A Viète-like formula for pi based on infinite sum of the arctangent functions with nested radicals</a>, figshare, 4509014, (2017).

%H Peter Bala, <a href="/A003881/a003881.pdf">Arctanh(z) and the Legendre polynomials</a>

%H Jonathan M. Borwein, Peter B. Borwein, and Karl Dilcher, <a href="http://www.jstor.org/stable/2324715">Pi, Euler numbers and asymptotic expansions</a>, Amer. Math. Monthly, 96 (1989), 681-687.

%H Ronald K. Hoeflin, <a href="https://web.archive.org/web/20140220050028/http://www.eskimo.com/~miyaguch/titan.html">Titan Test</a>.

%H Richard J. Mathar, <a href="https://arxiv.org/abs/1008.2547">Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli</a>, arXiv:1008.2547 [math.NT], 2010-2015.

%H Literate Programs, <a href="http://en.literateprograms.org/Pi_with_Machin&#39;s_formula_(Haskell)">Pi with Machin's formula (Haskell)</a>.

%H Michael Penn, <a href="https://www.youtube.com/watch?v=FG9tglvQrGo">A surprising appearance of pie!</a>, YouTube video, 2020.

%H Michael Penn, <a href="https://www.youtube.com/watch?v=PQ9vHEcIrU0">Transforming normal identities into "crazy" ones</a>, YouTube video, 2022.

%H Srinivasa Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/question/q353.htm">Question 353</a>, J. Ind. Math. Soc.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeProducts.html">Prime Products</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Leibniz_formula_for_π">Leibniz formula for Pi</a>.

%H <a href="/index/Go#GEB">Index entries for sequences from "Goedel, Escher, Bach"</a>.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.

%F Equals Integral_{x=0..oo} sin(2x)/(2x) dx.

%F Equals lim_{n->oo} n*A001586(n-1)/A001586(n) (conjecture). - _Mats Granvik_, Feb 23 2011

%F Equals Integral_{x=0..1} 1/(1+x^2) dx. - _Gary W. Adamson_, Jun 22 2003

%F 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

%F Equals (Sum_{x=0..oo} sin(x)*cos(x)/x) - 1/2. - _Bruno Berselli_, May 13 2013

%F Equals (-digamma(1/4) + digamma(3/4))/4. - _Jean-François Alcover_, May 31 2013

%F Equals Sum_{n>=0} (-1)^n/(2*n+1). - _Geoffrey Critzer_, Nov 03 2013

%F Equals Integral_{x=0..1} Product_{k>=1} (1-x^(8*k))^3 dx [cf. A258414]. - _Vaclav Kotesovec_, May 30 2015

%F Equals 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

%F From _Peter Bala_, Nov 15 2016: (Start)

%F 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).

%F 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.

%F 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.

%F 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).

%F 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)

%F From _Peter Bala_, Nov 05 2019: (Start)

%F For k = 0,1,2,..., Pi/4 = k!*Sum_{n = -oo..oo} 1/((4*n+1)*(4*n+3)* ...*(4*n+2*k+1)), where Sum_{n = -oo..oo} f(n) is understood as lim_{j -> oo} Sum_{n = -j..j} f(n).

%F Equals Integral_{x = 0..oo} sin(x)^4/x^2 dx = Sum_{n >= 1} sin(n)^4/n^2, by the Abel-Plana formula.

%F Equals Integral_{x = 0..oo} sin(x)^3/x dx = Sum_{n >= 1} sin(n)^3/n, by the Abel-Plana formula. (End)

%F From _Amiram Eldar_, Aug 19 2020: (Start)

%F Equals arcsin(1/sqrt(2)).

%F Equals Product_{k>=1} (1 - 1/(2*k+1)^2).

%F Equals Integral_{x=0..oo} x/(x^4 + 1) dx.

%F Equals Integral_{x=0..oo} 1/(x^2 + 4) dx. (End)

%F With offset 1, equals 5 * Pi / 2. - _Sean A. Irvine_, Aug 19 2021

%F Equals (1/2)!^2 = Gamma(3/2)^2. - _Gary W. Adamson_, Aug 23 2021

%F Equals Integral_{x = 0..oo} exp(-x)*sin(x)/x dx (see Rivaud reference). - _Bernard Schott_, Jan 28 2022

%F From _Amiram Eldar_, Nov 06 2023: (Start)

%F Equals beta(1), where beta is the Dirichlet beta function.

%F Equals Product_{p prime >= 3} (1 - (-1)^((p-1)/2)/p)^(-1). (End)

%F Equals arctan( F(1)/F(4) ) + arctan( F(2)/F(3) ), where F(1), F(2), F(3), and F(4) are any four consecutive Fibonacci numbers. - _Gary W. Adamson_, Mar 03 2024

%F Pi/4 = Sum_{n >= 1} i/(n*P(n, i)*P(n-1, i)) = (1/2)*Sum_{n >= 1} (-1)^(n+1)*4^n/(n*A006139(n)*A006139(n-1)), where i = sqrt(-1) and P(n, x) denotes the n-th Legendre polynomial. The n-th summand of the series is O( 1/(3 + 2*sqrt(3))^n ). - _Peter Bala_, Mar 16 2024

%F Equals arctan( phi^(-3) ) + arctan(phi^(-1) ). - _Gary W. Adamson_, Mar 27 2024

%F Equals Sum_{n>=1} eta(n)/2^n, where eta(n) is the Dirichlet eta function. - _Antonio Graciá Llorente_, Oct 04 2024

%e 0.785398163397448309615660845819875721049292349843776455243736148...

%e 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

%p evalf(Pi/4) ;

%t RealDigits[N[Pi/4,6! ]] (* _Vladimir Joseph Stephan Orlovsky_, Dec 02 2009 *)

%t (* PROGRAM STARTS *)

%t (* Define the nested radicals a_k by recurrence *)

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

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

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

%t N[Pi/4, 40]

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

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

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

%t (* Error terms for Pi/4 approximations *)

%t Print["Error terms for Pi/4"]

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

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

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

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

%t While[K <= 30,

%t AppendTo[sqn, {K,

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

%t N}]; K++]

%t Print[MatrixForm[sqn]]

%t (* _Sanjar Abrarov_, Jan 09 2017 *)

%o (Haskell) -- see link: Literate Programs

%o import Data.Char (digitToInt)

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

%o machin = 4 * arccot 5 unity - arccot 239 unity

%o unity = 10 ^ (len + 10)

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

%o arccot' x unity summa xpow n sign

%o | term == 0 = summa

%o | otherwise = arccot'

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

%o where term = xpow `div` n

%o -- _Reinhard Zumkeller_, Nov 20 2012

%o (SageMath) # Leibniz/Cohen/Villegas/Zagier/Arndt/Haenel

%o def FastLeibniz(n):

%o b = 2^(2*n-1); c = b; s = 0

%o for k in range(n-1,-1,-1):

%o t = 2*k+1

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

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

%o c += b

%o return s/c

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

%o print(A003881) # _Peter Luschny_, Nov 20 2012

%o (PARI) Pi/4 \\ _Charles R Greathouse IV_, Jul 07 2014

%o (Magma) R:= RealField(100); Pi(R)/4; // _G. C. Greubel_, Mar 08 2018

%Y Cf. A000796, A001586, A071904, A019669, A197723, A347152.

%Y Cf. A000182, A000364, A024235, A278080, A278195.

%Y Cf. A006752 (beta(2)=Catalan), A153071 (beta(3)), A175572 (beta(4)), A175571 (beta(5)), A175570 (beta(6)), A258814 (beta(7)), A258815 (beta(8)), A258816 (beta(9)).

%Y Cf. A001622.

%K nonn,cons,easy

%O 0,1

%A _N. J. A. Sloane_, _Simon Plouffe_

%E a(98) and a(99) corrected by _Reinhard Zumkeller_, Nov 20 2012