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A003881 Decimal expansion of Pi/4.

%I A003881 #225 Aug 07 2025 17:40:50
%S A003881 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 A003881 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 A003881 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 A003881 Decimal expansion of Pi/4.
%C A003881 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 A003881 Also the surface area of a quarter-sphere of diameter 1. - _Omar E. Pol_, Oct 03 2013
%C A003881 Least positive solution to sin(x) = cos(x). - _Franklin T. Adams-Watters_, Jun 17 2014
%C A003881 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 A003881 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 A003881 Pi/4 is the area enclosed between circumcircle and incircle of a regular polygon of unit side. - _Mohammed Yaseen_, Nov 29 2023
%D A003881 Jörg Arndt and Christoph Haenel, Pi: Algorithmen, Computer, Arithmetik, Springer 2000, p. 150.
%D A003881 Florian Cajori, A History of Mathematical Notations, Dover edition (2012), par. 437.
%D A003881 Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 6.3 and 8.4, pp. 429 and 492.
%D A003881 Douglas R. Hofstadter, Gödel, Escher, Bach: An Eternal Golden Braid, Basic Books, p. 408.
%D A003881 J. Rivaud, Analyse, Séries, équations différentielles, Mathématiques supérieures et spéciales, Premier cycle universitaire, Vuibert, 1981, Exercice 3, p. 136.
%D A003881 James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 119.
%H A003881 Reinhard Zumkeller, <a href="/A003881/b003881.txt">Table of n, a(n) for n = 0..1000</a>
%H A003881 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 A003881 Peter Bala, <a href="/A003881/a003881.pdf">Arctanh(z) and the Legendre polynomials</a>
%H A003881 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 A003881 Antonio Gracia Llorente, <a href="https://osf.io/preprints/osf/hj5zp_v1">Shifting Constants Through Infinite Product Transformations</a>, OSF Preprint, 2024.
%H A003881 Ronald K. Hoeflin, <a href="https://web.archive.org/web/20140220050028/http://www.eskimo.com/~miyaguch/titan.html">Titan Test</a>.
%H A003881 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 A003881 Literate Programs, <a href="http://en.literateprograms.org/Pi_with_Machin&#39;s_formula_(Haskell)">Pi with Machin's formula (Haskell)</a>.
%H A003881 Michael Penn, <a href="https://www.youtube.com/watch?v=FG9tglvQrGo">A surprising appearance of pie!</a>, YouTube video, 2020.
%H A003881 Michael Penn, <a href="https://www.youtube.com/watch?v=PQ9vHEcIrU0">Transforming normal identities into "crazy" ones</a>, YouTube video, 2022.
%H A003881 Srinivasa Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/question/q353.htm">Question 353</a>, J. Ind. Math. Soc.
%H A003881 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeProducts.html">Prime Products</a>.
%H A003881 Wikipedia, <a href="https://en.wikipedia.org/wiki/Leibniz_formula_for_π">Leibniz formula for Pi</a>.
%H A003881 <a href="/index/Go#GEB">Index entries for sequences from "Goedel, Escher, Bach"</a>.
%H A003881 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F A003881 Equals Integral_{x=0..oo} sin(2x)/(2x) dx.
%F A003881 Equals lim_{n->oo} n*A001586(n-1)/A001586(n) (conjecture). - _Mats Granvik_, Feb 23 2011
%F A003881 Equals Integral_{x=0..1} 1/(1+x^2) dx. - _Gary W. Adamson_, Jun 22 2003
%F A003881 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 A003881 Equals (Sum_{x=0..oo} sin(x)*cos(x)/x) - 1/2. - _Bruno Berselli_, May 13 2013
%F A003881 Equals (-digamma(1/4) + digamma(3/4))/4. - _Jean-François Alcover_, May 31 2013
%F A003881 Equals Sum_{n>=0} (-1)^n/(2*n+1). - _Geoffrey Critzer_, Nov 03 2013
%F A003881 Equals Integral_{x=0..1} Product_{k>=1} (1-x^(8*k))^3 dx [cf. A258414]. - _Vaclav Kotesovec_, May 30 2015
%F A003881 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 A003881 From _Peter Bala_, Nov 15 2016: (Start)
%F A003881 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 A003881 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 A003881 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 A003881 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 A003881 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 A003881 From _Peter Bala_, Nov 05 2019: (Start)
%F A003881 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 A003881 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 A003881 Equals Integral_{x = 0..oo} sin(x)^3/x dx = Sum_{n >= 1} sin(n)^3/n, by the Abel-Plana formula. (End)
%F A003881 From _Amiram Eldar_, Aug 19 2020: (Start)
%F A003881 Equals arcsin(1/sqrt(2)).
%F A003881 Equals Product_{k>=1} (1 - 1/(2*k+1)^2).
%F A003881 Equals Integral_{x=0..oo} x/(x^4 + 1) dx.
%F A003881 Equals Integral_{x=0..oo} 1/(x^2 + 4) dx. (End)
%F A003881 With offset 1, equals 5 * Pi / 2. - _Sean A. Irvine_, Aug 19 2021
%F A003881 Equals (1/2)!^2 = Gamma(3/2)^2. - _Gary W. Adamson_, Aug 23 2021
%F A003881 Equals Integral_{x = 0..oo} exp(-x)*sin(x)/x dx (see Rivaud reference). - _Bernard Schott_, Jan 28 2022
%F A003881 From _Amiram Eldar_, Nov 06 2023: (Start)
%F A003881 Equals beta(1), where beta is the Dirichlet beta function.
%F A003881 Equals Product_{p prime >= 3} (1 - (-1)^((p-1)/2)/p)^(-1). (End)
%F A003881 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 A003881 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 A003881 Equals arctan( phi^(-3) ) + arctan(phi^(-1) ). - _Gary W. Adamson_, Mar 27 2024
%F A003881 Equals Sum_{n>=1} eta(n)/2^n, where eta(n) is the Dirichlet eta function. - _Antonio Graciá Llorente_, Oct 04 2024
%F A003881 Equals Product_{k>=2} ((k + 1)^(k*(2*k + 1))*(k - 1)^(k*(2*k - 1)))/k^(4*k^2). - _Antonio Graciá Llorente_, Apr 12 2025
%F A003881 Equals Integral_{x=sqrt(2)..oo} dx/(x*sqrt(x^2 - 1)). - _Kritsada Moomuang_, May 29 2025
%e A003881 0.785398163397448309615660845819875721049292349843776455243736148...
%e A003881 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 A003881 evalf(Pi/4) ;
%t A003881 RealDigits[N[Pi/4,6! ]]  (* _Vladimir Joseph Stephan Orlovsky_, Dec 02 2009 *)
%t A003881 (* PROGRAM STARTS *)
%t A003881 (* Define the nested radicals a_k by recurrence *)
%t A003881 a[k_] := Nest[Sqrt[2 + #1] & , 0, k]
%t A003881 (* Example of Pi/4 approximation at K = 100 *)
%t A003881 Print["The actual value of Pi/4 is"]
%t A003881 N[Pi/4, 40]
%t A003881 Print["At K = 100 the approximated value of Pi/4 is"]
%t A003881 K := 100;  (* the truncating integer *)
%t A003881 N[Sum[ArcTan[Sqrt[2 - a[k - 1]]/a[k]], {k, 2, K}], 40] (* equation (8) *)
%t A003881 (* Error terms for Pi/4 approximations *)
%t A003881 Print["Error terms for Pi/4"]
%t A003881 k := 1; (* initial value of the index k *)
%t A003881 K := 10; (* initial value of the truncating integer K *)
%t A003881 sqn := {}; (* initiate the sequence *)
%t A003881 AppendTo[sqn, {"Truncating integer K ", " Error term in Pi/4"}];
%t A003881 While[K <= 30,
%t A003881 AppendTo[sqn, {K,
%t A003881    N[Pi/4 - Sum[ArcTan[Sqrt[2 - a[k - 1]]/a[k]], {k, 2, K}], 1000] //
%t A003881     N}]; K++]
%t A003881 Print[MatrixForm[sqn]]
%t A003881 (* _Sanjar Abrarov_, Jan 09 2017 *)
%o A003881 (Haskell) -- see link: Literate Programs
%o A003881 import Data.Char (digitToInt)
%o A003881 a003881_list len = map digitToInt $ show $ machin `div` (10 ^ 10) where
%o A003881    machin = 4 * arccot 5 unity - arccot 239 unity
%o A003881    unity = 10 ^ (len + 10)
%o A003881    arccot x unity = arccot' x unity 0 (unity `div` x) 1 1 where
%o A003881      arccot' x unity summa xpow n sign
%o A003881     | term == 0 = summa
%o A003881     | otherwise = arccot'
%o A003881       x unity (summa + sign * term) (xpow `div` x ^ 2) (n + 2) (- sign)
%o A003881     where term = xpow `div` n
%o A003881 -- _Reinhard Zumkeller_, Nov 20 2012
%o A003881 (SageMath) # Leibniz/Cohen/Villegas/Zagier/Arndt/Haenel
%o A003881 def FastLeibniz(n):
%o A003881     b = 2^(2*n-1); c = b; s = 0
%o A003881     for k in range(n-1,-1,-1):
%o A003881         t = 2*k+1
%o A003881         s = s + c/t if is_even(k) else s - c/t
%o A003881         b *= (t*(k+1))/(2*(n-k)*(n+k))
%o A003881         c += b
%o A003881     return s/c
%o A003881 A003881 = RealField(3333)(FastLeibniz(1330))
%o A003881 print(A003881)  # _Peter Luschny_, Nov 20 2012
%o A003881 (PARI) Pi/4 \\ _Charles R Greathouse IV_, Jul 07 2014
%o A003881 (Magma) R:= RealField(100); Pi(R)/4; // _G. C. Greubel_, Mar 08 2018
%Y A003881 Cf. A000796, A001586, A071904, A019669, A197723, A347152.
%Y A003881 Cf. A000182, A000364, A024235, A278080, A278195.
%Y A003881 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 A003881 Cf. A001622.
%K A003881 nonn,cons,easy
%O A003881 0,1
%A A003881 _N. J. A. Sloane_, _Simon Plouffe_
%E A003881 a(98) and a(99) corrected by _Reinhard Zumkeller_, Nov 20 2012