A092425 -id:A092425 - cp's OEIS frontend

A000796 Decimal expansion of Pi (or digits of Pi).

3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 8, 9, 7, 9, 3, 2, 3, 8, 4, 6, 2, 6, 4, 3, 3, 8, 3, 2, 7, 9, 5, 0, 2, 8, 8, 4, 1, 9, 7, 1, 6, 9, 3, 9, 9, 3, 7, 5, 1, 0, 5, 8, 2, 0, 9, 7, 4, 9, 4, 4, 5, 9, 2, 3, 0, 7, 8, 1, 6, 4, 0, 6, 2, 8, 6, 2, 0, 8, 9, 9, 8, 6, 2, 8, 0, 3, 4, 8, 2, 5, 3, 4, 2, 1, 1, 7, 0, 6, 7, 9, 8, 2, 1, 4

Offset: 1

N. J. A. Sloane

Sometimes called Archimedes's constant.
Ratio of a circle's circumference to its diameter.
Also area of a circle with radius 1.
Also surface area of a sphere with diameter 1.
A useful mnemonic for remembering the first few terms: How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics ...
Also ratio of surface area of sphere to one of the faces of the circumscribed cube. Also ratio of volume of a sphere to one of the six inscribed pyramids in the circumscribed cube. - Omar E. Pol, Aug 09 2012
Also surface area of a quarter of a sphere of radius 1. - Omar E. Pol, Oct 03 2013
Also the area under the peak-shaped even function f(x)=1/cosh(x). Proof: for the upper half of the integral, write f(x) = (2*exp(-x))/(1+exp(-2x)) = 2*Sum_{k>=0} (-1)^k*exp(-(2k+1)*x) and integrate term by term from zero to infinity. The result is twice the Gregory series for Pi/4. - Stanislav Sykora, Oct 31 2013
A curiosity: a 144 X 144 magic square of 7th powers was recently constructed by Toshihiro Shirakawa. The magic sum = 3141592653589793238462643383279502884197169399375105, which is the concatenation of the first 52 digits of Pi. See the MultiMagic Squares link for details. - Christian Boyer, Dec 13 2013 [Comment revised by N. J. A. Sloane, Aug 27 2014]
x*Pi is also the surface area of a sphere whose diameter equals the square root of x. - Omar E. Pol, Dec 25 2013
Also diameter of a sphere whose surface area equals the volume of the circumscribed cube. - Omar E. Pol, Jan 13 2014
From Daniel Forgues, Mar 20 2015: (Start)
An interesting anecdote about the base-10 representation of Pi, with 3 (integer part) as first (index 1) digit:
  358 0
  359 3
  360 6
  361 0
  362 0
And the circle is customarily subdivided into 360 degrees (although Pi radians yields half the circle)...
(End)
Sometimes referred to as Archimedes's constant, because the Greek mathematician computed lower and upper bounds of Pi by drawing regular polygons inside and outside a circle. In Germany it was called the Ludolphian number until the early 20th century after the Dutch mathematician Ludolph van Ceulen (1540-1610), who calculated up to 35 digits of Pi in the late 16th century. - Martin Renner, Sep 07 2016
As of the beginning of 2019 more than 22 trillion decimal digits of Pi are known. See the Wikipedia article "Chronology of computation of Pi". - Harvey P. Dale, Jan 23 2019
On March 14, 2019, Emma Haruka Iwao announced the calculation of 31.4 trillion digits of Pi using Google Cloud's infrastructure. - David Radcliffe, Apr 10 2019
Also volume of three quarters of a sphere of radius 1. - Omar E. Pol, Aug 16 2019
On August 5, 2021, researchers from the University of Applied Sciences of the Grisons in Switzerland announced they had calculated 62.8 trillion digits. Guinness World Records has not verified this yet. - Alonso del Arte, Aug 23 2021
The Hermite-Lindemann (1882) theorem states, that if z is a nonzero algebraic number, then e^z is a transcendent number. The transcendence of Pi then results from Euler's relation: e^(i*Pi) = -1. - Peter Luschny, Jul 21 2023
The 10000 words of the book "Not A Wake" by Michael Keith, written in Pilish, match in length the first 10000 digits of Pi. - Paolo Xausa, Aug 07 2025

			3.1415926535897932384626433832795028841971693993751058209749445923078164062\
862089986280348253421170679821480865132823066470938446095505822317253594081\
284811174502841027019385211055596446229489549303819...

Mohammad K. Azarian, A Summary of Mathematical Works of Ghiyath ud-din Jamshid Kashani, Journal of Recreational Mathematics, Vol. 29(1), pp. 32-42, 1998.
J. Arndt & C. Haenel, Pi Unleashed, Springer NY 2001.
P. Beckmann, A History of Pi, Golem Press, Boulder, CO, 1977.
Florian Cajori, A History of Mathematical Notations, Dover edition (2012), par. 396.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 24, 237-239.
J.-P. Delahaye, Le fascinant nombre pi, Pour la Science, Paris 1997.
P. Eyard and J.-P. Lafon, The Number Pi, Amer. Math. Soc., 2004.
S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, Section 1.4.
Le Petit Archimede, Special Issue On Pi, Supplement to No. 64-5, May 1980 ADCS Amiens.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 31.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 1, equations 1:7:1, 1:7:2 at pages 12-13.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See pp. 48-55.

Harry J. Smith, Table of n, a(n) for n = 1..20000
Sanjar M. Abrarov, Rehan Siddiqui, Rajinder K. Jagpal, and Brendan M. Quine, Unconditional applicability of the Lehmer's measure to the two-term Machin-like formula for pi, arXiv:2004.11711 [math.GM], 2020.
Emilio Ambrisi and Bruno Rizzi, Appunti da un corso di aggiornamento, Mathesis (Sezione Casertana), Quaderno n. 1, Liceo G. Galilei, Mondragone (CE), Italy, June 22-28 1979. (In Italian). See p. 15.
Dave Andersen, Pi-Search Page
Anonymous, A million digits of Pi
Anonymous, Liste de quelques milliers de decimales du nombre de pi
D. H. Bailey, On Kanada's computation of 1.24 trillion digits of Pi [archived page]
D. H. Bailey and J. M. Borwein, Experimental Mathematics: Examples, Methods and Implications, Notices of the AMS, Volume 52, Number 5, May 2005, pp. 502-514.
Harry Baker, "Pi calculated to a record-breaking 62.8 trillion digits", Live Science, August 17, 2021.
Steve Baker and Thomas Moore, 100 trillion digits of pi
Frits Beukers, A rational approach to Pi, Nieuw Archief voor de Wiskunde, December 2000, pp. 372-379.
J. M. Borwein, Talking about Pi
J. M. Borwein and M. Macklem, The (Digital) Life of Pi, The Australian Mathematical Society Gazette, Volume 33, Number 5, Sept. 2006, pp. 243-248.
Peter Borwein, The amazing number Pi, Nieuw Archief voor de Wiskunde, September 2000, pp. 254-258.
Christian Boyer, MultiMagic Squares
J. Britton, Mnemonics For The Number Pi [archived page]
D. Castellanos, The ubiquitous pi, Math. Mag., 61 (1988), 67-98 and 148-163.
Jonas Castillo Toloza, Fascinating Method for Finding Pi
E. S. Croot, Pade Approximations and the Transcendence of pi
L. Euler, On the sums of series of reciprocals, arXiv:math/0506415 [math.HO], 2005-2008.
L. Euler, De summis serierum reciprocarum, E41.
Eureka, Tout pi or not tout pi
Ph. Flajolet and I. Vardi, Zeta function expansions of some classical constants
Jeremy Gibbons, Unbounded Spigot Algorithms for the Digits of Pi
GJ, 10 million digits of Pi
X. Gourdon, Pi to 16000 decimals [archived page]
Xavier Gourdon, A new algorithm for computing Pi in base 10
X. Gourdon and P. Sebah, Archimedes' constant Pi
B. Gourevitch, L'univers de Pi
Antonio Gracia Llorente, Novel Infinite Products πe and π/e, OSF Preprint, 2024.
L. Grebelius, Approximation of Pi: First 1000000 digits
J. Guillera and J. Sondow, Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent, Ramanujan J. 16 (2008) 247-270. Preprint: arXiv:math/0506319 [math.NT] (2005-2006).
Carl-Johan Haster, Pi from the sky -- A null test of general relativity from a population of gravitational wave observations, arXiv:2005.05472 [gr-qc], 2020.
H. Havermann, Simple Continued Fraction for Pi [archived page]
M. D. Huberty et al., 100000 Digits of Pi
ICON Project, Pi to 50000 places [archived page]
Emma Haruka Iwao, Pi in the sky: Calculating a record-breaking 31.4 trillion digits of Archimedes' constant on Google Cloud
P. Johns, 120000 Digits of Pi [archived page]
Yasumasa Kanada, 1.24 trillion digits of Pi
Yasumasa Kanada and Daisuke Takahashi, 206 billion digits of Pi [archived page]
Literate Programs, Pi with Machin's formula (Haskell) [archived page]
Johannes W. Meijer, Pi everywhere poster, Mar 14 2013
J. Moyer, First 10000 digits of pi
NERSC, Search Pi [broken link]
Remco Niemeijer, The Digits of Pi, programmingpraxis.
Steve Pagliarulo, Stu's pi page [archived page]
Chittaranjan Pardeshi, BBP-Like formula for Pi in Golden Ratio Base Phi
Michael Penn, A nice inverse tangent integral., YouTube video, 2020.
Michael Penn, Pi is irrational (π∉ℚ), YouTube video, 2020.
I. Peterson, A Passion for Pi
G. M. Phillips, Table of contents of "Pi: A source Book"
Simon Plouffe, 10000 digits of Pi
Simon Plouffe, A formula for the nth decimal digit or binary of Pi and powers of Pi, arXiv:2201.12601 [math.NT], 2022.
D. Pothet, Chronologie du calcul des decimales de pi [broken link]
M. Z. Rafat and D. Dobie, Throwing Pi at a wall, arXiv:1901.06260 [physics.class-ph], 2020.
S. Ramanujan, Modular equations and approximations to \pi, Quart. J. Math. 45 (1914), 350-372.
H. Ricardo, Review of "The Number Pi" by P. Eymard & J.-P. Lafon
M. Ripa and G. Morelli, Retro-analytical Reasoning IQ tests for the High Range, 2013.
Grant Sanderson, Why do colliding blocks compute pi?, 3Blue1Brown video (2019).
Daniel B. Sedory, The Pi Pages
Michael I. Shamos, A catalog of the real numbers, (2007). See p. 180.
D. Shanks and J. W. Wrench, Jr., Calculation of pi to 100,000 decimals, Math. Comp. 16 1962 76-99.
Jean-Louis Sigrist, Les 128000 premieres decimales du nombre PI
Sizes, pi
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 5.
Katie Steckles, Pi and constrained writing, The Aperiodical, 2015.
A. Sofo, Pi and some other constants, Journal of Inequalities in Pure and Applied Mathematics, Vol. 6, Issue 5, Article 138, 2005.
Jonathan Sondow, A faster product for Pi and a new integral for ln Pi/2, arXiv:math/0401406 [math.NT], 2004; Amer. Math. Monthly 112 (2005) 729-734.
D. Surendran, Can I have a small container of coffee? [archived page]
Wislawa Szymborska, Pi (The admirable number Pi), Miracle Fair, 2002.
G. Vacca, A new analytical expression for the number pi, and some historical considerations, Bull. Amer. Math. Soc. 16 (1910), 368-369.
Stan Wagon, Is Pi Normal?
Eric Weisstein's World of Mathematics, Pi and Pi Digits
Wikipedia, Bailey-Borwein-Plouffe formula, Normal Number, Pi, and Machin-like formula
Wikipedia, Pilish.
Alexander J. Yee & Shigeru Kondo, 5 Trillion Digits of Pi - New World Record
Alexander J. Yee & Shigeru Kondo, Round 2... 10 Trillion Digits of Pi
Index entries for sequences related to the number Pi
Index entries for "core" sequences
Index entries for transcendental numbers

Cf. A001203 (continued fraction).
Pi in base b: A004601 (b=2), A004602 (b=3), A004603 (b=4), A004604 (b=5), A004605 (b=6), A004606 (b=7), A006941 (b=8), A004608 (b=9), this sequence (b=10), A068436 (b=11), A068437 (b=12), A068438 (b=13), A068439 (b=14), A068440 (b=15), A062964 (b=16), A224750 (b=26), A224751 (b=27), A060707 (b=60). - Jason Kimberley, Dec 06 2012
Decimal expansions of expressions involving Pi: A002388 (Pi^2), A003881 (Pi/4), A013661 (Pi^2/6), A019692 (2*Pi=tau), A019727 (sqrt(2*Pi)), A059956 (6/Pi^2), A060294 (2/Pi), A091925 (Pi^3), A092425 (Pi^4), A092731 (Pi^5), A092732 (Pi^6), A092735 (Pi^7), A092736 (Pi^8), A163973 (Pi/log(2)).
Cf. A001901 (Pi/2; Wallis), A002736 (Pi^2/18; Euler), A007514 (Pi), A048581 (Pi; BBP), A054387 (Pi; Newton), A092798 (Pi/2), A096954 (Pi/4; Machin), A097486 (Pi), A122214 (Pi/2), A133766 (Pi/4 - 1/2), A133767 (5/6 - Pi/4), A166107 (Pi; MGL).
Cf. A001622, A048940, A248682.

-- see link: Literate Programs
import Data.Char (digitToInt)
a000796 n = a000796_list (n + 1) !! (n + 1)
a000796_list len = map digitToInt $ show $ machin' `div` (10 ^ 10) where
   machin' = 4 * (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 24 2012

-- See Niemeijer link and also Gibbons link.
a000796 n = a000796_list !! (n-1) :: Int
a000796_list = map fromInteger $ piStream (1, 0, 1)
   [(n, a*d, d) | (n, d, a) <- map (\k -> (k, 2 * k + 1, 2)) [1..]] where
   piStream z xs'@(x:xs)
     | lb /= approx z 4 = piStream (mult z x) xs
     | otherwise = lb : piStream (mult (10, -10 * lb, 1) z) xs'
     where lb = approx z 3
           approx (a, b, c) n = div (a * n + b) c
           mult (a, b, c) (d, e, f) = (a * d, a * e + b * f, c * f)
-- Reinhard Zumkeller, Jul 14 2013, Jun 12 2013

py(x) := if equal(6,6+x^2) then 2*x else (py(x:x/3),3*%%-4*(%%-x)^3); py(3.); py(dfloat(%)); block([bfprecision:35], py(bfloat(%))) /* Bill Gosper, Sep 09 2002 */

pi:=Pi(RealField(110)); Reverse(Intseq(Floor(10^105*pi))); // Bruno Berselli, Mar 12 2013

Digits := 110: Pi*10^104:
ListTools:-Reverse(convert(floor(%), base, 10)); # Peter Luschny, Oct 29 2019

RealDigits[ N[ Pi, 105]] [[1]]
Table[ResourceFunction["NthDigit"][Pi, n], {n, 1, 102}] (* Joan Ludevid, Jun 22 2022; easy to compute a(10000000)=7 with this function; requires Mathematica 12.0+ *)

{ default(realprecision, 20080); x=Pi; for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b000796.txt", n, " ", d)); } \\ Harry J. Smith, Apr 15 2009

A796=[]; A000796(n)={if(n>#A796, localprec(n*6\5+29); A796=digits(Pi\.1^(precision(Pi)-3))); A796[n]} \\ NOTE: as the other programs, this returns the n-th term of the sequence, with n = 1, 2, 3, ... and not n = 1, 0, -1, -2, .... - M. F. Hasler, Jun 21 2022

first(n)= default(realprecision, n+10); digits(floor(Pi*10^(n-1))) \\ David A. Corneth, Jun 21 2022

lista(nn, p=20)= {my(u=10^(nn+p+1), f(x, u)=my(n=1, q=u\x, r=q, s=1, t); while(t=(q\=(x*x))\(n+=2), r+=(s=-s)*t); r*4); digits((4*f(5, u)-f(239, u))\10^(p+2)); } \\ Machin-like, with p > the maximal number of consecutive 9-digits to be expected (A048940) - Ruud H.G. van Tol, Dec 26 2024

from sympy import pi, N; print(N(pi, 1000)) # David Radcliffe, Apr 10 2019

from mpmath import mp
def A000796(n):
    if n >= len(A000796.str): mp.dps = n*6//5+50; A000796.str = str(mp.pi-5/mp.mpf(10)**mp.dps)
    return int(A000796.str[n if n>1 else 0])
A000796.str = '' # M. F. Hasler, Jun 21 2022

m=125
x=numerical_approx(pi, digits=m+5)
a=[ZZ(i) for i in x.str(skip_zeroes=True) if i.isdigit()]
a[:m] # G. C. Greubel, Jul 18 2023

Pi = 4*Sum_{k>=0} (-1)^k/(2k+1) [Madhava-Gregory-Leibniz, 1450-1671]. - N. J. A. Sloane, Feb 27 2013
From Johannes W. Meijer, Mar 10 2013: (Start)
2/Pi = (sqrt(2)/2) * (sqrt(2 + sqrt(2))/2) * (sqrt(2 + sqrt(2 + sqrt(2)))/2) * ... [Viete, 1593]
2/Pi = Product_{k>=1} (4*k^2-1)/(4*k^2). [Wallis, 1655]
Pi = 3*sqrt(3)/4 + 24*(1/12 - Sum_{n>=2} (2*n-2)!/((n-1)!^2*(2*n-3)*(2*n+1)*2^(4*n-2))). [Newton, 1666]
Pi/4 = 4*arctan(1/5) - arctan(1/239). [Machin, 1706]
Pi^2/6 = 3*Sum_{n>=1} 1/(n^2*binomial(2*n,n)). [Euler, 1748]
1/Pi = (2*sqrt(2)/9801) * Sum_{n>=0} (4*n)!*(1103+26390*n)/((n!)^4*396^(4*n)). [Ramanujan, 1914]
1/Pi = 12*Sum_{n>=0} (-1)^n*(6*n)!*(13591409 + 545140134*n)/((3*n)!*(n!)^3*(640320^3)^(n+1/2)). [David and Gregory Chudnovsky, 1989]
Pi = Sum_{n>=0} (1/16^n) * (4/(8*n+1) - 2/(8*n+4) - 1/(8*n+5) - 1/(8*n+6)). [Bailey-Borwein-Plouffe, 1989] (End)
Pi = 4 * Sum_{k>=0} 1/(4*k+1) - 1/(4*k+3). - Alexander R. Povolotsky, Dec 25 2008
Pi = 4*sqrt(-1*(Sum_{n>=0} (i^(2*n+1))/(2*n+1))^2). - Alexander R. Povolotsky, Jan 25 2009
Pi = Integral_{x=-oo..oo} dx/(1+x^2). - Mats Granvik and Gary W. Adamson, Sep 23 2012
Pi - 2 = 1/1 + 1/3 - 1/6 - 1/10 + 1/15 + 1/21 - 1/28 - 1/36 + 1/45 + ... [Jonas Castillo Toloza, 2007], that is, Pi - 2 = Sum_{n>=1} (1/((-1)^floor((n-1)/2)*(n^2+n)/2)). - José de Jesús Camacho Medina, Jan 20 2014
Pi = 3 * Product_{t=img(r),r=(1/2+i*t) root of zeta function} (9+4*t^2)/(1+4*t^2) <=> RH is true. - Dimitris Valianatos, May 05 2016
From Ilya Gutkovskiy, Aug 07 2016: (Start)
Pi = Sum_{k>=1} (3^k - 1)*zeta(k+1)/4^k.
Pi = 2*Product_{k>=2} sec(Pi/2^k).
Pi = 2*Integral_{x>=0} sin(x)/x dx. (End)
Pi = 2^{k + 1}*arctan(sqrt(2 - a_{k - 1})/a_k) at k >= 2, where a_k = sqrt(2 + a_{k - 1}) and a_1 = sqrt(2). - Sanjar Abrarov, Feb 07 2017
Pi = Integral_{x = 0..2} sqrt(x/(2 - x)) dx. - Arkadiusz Wesolowski, Nov 20 2017
Pi = lim_{n->oo} 2/n * Sum_{m=1,n} ( sqrt( (n+1)^2 - m^2 ) - sqrt( n^2 - m^2 ) ). - Dimitri Papadopoulos, May 31 2019
From Peter Bala, Oct 29 2019: (Start)
Pi = Sum_{n >= 0} 2^(n+1)/( binomial(2*n,n)*(2*n + 1) ) - Euler.
More generally, Pi = (4^x)*x!/(2*x)! * Sum_{n >= 0} 2^(n+1)*(n+x)!*(n+2*x)!/(2*n+2*x+1)! = 2*4^x*x!^2/(2*x+1)! * hypergeom([2*x+1,1], [x+3/2], 1/2), valid for complex x not in {-1,-3/2,-2,-5/2,...}. Here, x! is shorthand notation for the function Gamma(x+1). This identity may be proved using Gauss's second summation theorem.
Setting x = 3/4 and x = -1/4 (resp. x = 1/4 and x = -3/4) in the above identity leads to series representations for the constant A085565 (resp. A076390). (End)
Pi = Im(log(-i^i)) = log(i^i)*(-2). - Peter Luschny, Oct 29 2019
From Amiram Eldar, Aug 15 2020: (Start)
Equals 2 + Integral_{x=0..1} arccos(x)^2 dx.
Equals Integral_{x=0..oo} log(1 + 1/x^2) dx.
Equals Integral_{x=0..oo} log(1 + x^2)/x^2 dx.
Equals Integral_{x=-oo..oo} exp(x/2)/(exp(x) + 1) dx. (End)
Equals 4*(1/2)!^2 = 4*Gamma(3/2)^2. - Gary W. Adamson, Aug 23 2021
From Peter Bala, Dec 08 2021: (Start)
Pi = 32*Sum_{n >= 1} (-1)^n*n^2/((4*n^2 - 1)*(4*n^2 - 9))= 384*Sum_{n >= 1} (-1)^(n+1)*n^2/((4*n^2 - 1)*(4*n^2 - 9)*(4*n^2 - 25)).
More generally, it appears that for k = 1,2,3,..., Pi = 16*(2*k)!*Sum_{n >= 1} (-1)^(n+k+1)*n^2/((4*n^2 - 1)* ... *(4*n^2 - (2*k+1)^2)).
Pi = 32*Sum_{n >= 1} (-1)^(n+1)*n^2/(4*n^2 - 1)^2 = 768*Sum_{n >= 1} (-1)^(n+1)*n^2/((4*n^2 - 1)^2*(4*n^2 - 9)^2).
More generally, it appears that for k = 0,1,2,..., Pi = 16*Catalan(k)*(2*k)!*(2*k+2)!*Sum_{n >= 1} (-1)^(n+1)*n^2/((4*n^2 - 1)^2* ... *(4*n^2 - (2*k+1)^2)^2).
Pi = (2^8)*Sum_{n >= 1} (-1)^(n+1)*n^2/(4*n^2 - 1)^4 = (2^17)*(3^5)*Sum_{n >= 2} (-1)^n*n^2*(n^2 - 1)/((4*n^2 - 1)^4*(4*n^2 - 9)^4) = (2^27)*(3^5)*(5^5)* Sum_{n >= 3} (-1)^(n+1)*n^2*(n^2 - 1)*(n^2 - 4)/((4*n^2 - 1)^4*(4*n^2 - 9)^4*(4*n^2 - 25)^4). (End)
For odd n, Pi = (2^(n-1)/A001818((n-1)/2))*gamma(n/2)^2. - Alan Michael Gómez Calderón, Mar 11 2022
Pi = 4/phi + Sum_{n >= 0} (1/phi^(12*n)) * ( 8/((12*n+3)*phi^3) + 4/((12*n+5)*phi^5) - 4/((12*n+7)*phi^7) - 8/((12*n+9)*phi^9) - 4/((12*n+11)*phi^11) + 4/((12*n+13)*phi^13) ) where phi = (1+sqrt(5))/2. - Chittaranjan Pardeshi, May 16 2022
Pi = sqrt(3)*(27*S - 36)/24, where S = A248682. - Peter Luschny, Jul 22 2022
Equals Integral_{x=0..1} 1/sqrt(x-x^2) dx. - Michal Paulovic, Sep 24 2023
From Peter Bala, Oct 28 2023: (Start)
Pi = 48*Sum_{n >= 0} (-1)^n/((6*n + 1)*(6*n + 3)*(6*n + 5)).
More generally, it appears that for k >= 0 we have Pi = A(k) + B(k)*Sum_{n >= 0} (-1)^n/((6*n + 1)*(6*n + 3)*...*(6*n + 6*k + 5)), where A(k) is a rational approximation to Pi and B(k) = (3 * 2^(3*k+3) * (3*k + 2)!) / (2^(3*k+1) - (-1)^k). The first few values of A(k) for k >= 0 are [0, 256/85, 65536/20955, 821559296/261636375, 6308233216/2008080987, 908209489444864/289093830828075, ...].
Pi = 16/5 - (288/5)*Sum_{n >= 0} (-1)^n * (6*n + 1)/((6*n + 1)*(6*n + 3)*...*(6*n + 9)).
More generally, it appears that for k >= 0 we have Pi = C(k) + D(k)*Sum_{n >= 0} (-1)^n* (6*n + 1)/((6*n + 1)*(6*n + 3)*...*(6*n + 6*k + 3)), where C(k) and D(k) are rational numbers. The case k = 0 is the Madhava-Gregory-Leibniz series for Pi.
Pi = 168/53 + (288/53)*Sum_{n >= 0} (-1)^n * (42*n^2 + 25*n)/((6*n + 1)*(6*n + 3)*(6*n + 5)*(6*n + 7)).
More generally, it appears that for k >= 1 we have Pi = E(k) + F(k)*Sum_{n >= 0} (-1)^n * (6*(6*k + 1)*n^2 + (24*k + 1)*n)/((6*n + 1)*(6*n + 3)*...*(6*n + 6*k + 1)), where E(k) and F(k) are rational numbers. (End)
From Peter Bala, Nov 10 2023: (Start)
The series representation Pi = 4 * Sum_{k >= 0} 1/(4*k + 1) - 1/(4*k + 3) given above by Alexander R. Povolotsky, Dec 25 2008, is the case n = 0 of the more general result (obtained by the WZ method): for n >= 0, there holds
Pi = Sum_{j = 0.. n-1} 2^(j+1)/((2*j + 1)*binomial(2*j,j)) + 8*(n+1)!*Sum_{k >= 0} 1/((4*k + 1)*(4*k + 3)*...*(4*k + 2*n + 3)).
Letting n -> oo gives the rapidly converging series Pi = Sum_{j >= 0} 2^(j+1)/((2*j + 1)*binomial(2*j,j)) due to Euler.
More generally, it appears that for n >= 1, Pi = 1/(2*n-1)!!^2 * Sum_{j >= 0} (Product_{i = 0..2*n-1} j - i) * 2^(j+1)/((2*j + 1)*binomial(2*j,j)).
For any integer n, Pi = (-1)^n * 4 * Sum_{k >= 0} 1/(4*k + 1 + 2*n) - 1/(4*k + 3 - 2*n). (End)
Pi = Product_{k>=1} ((k^3*(k + 2)*(2*k + 1)^2)/((k + 1)^4*(2*k - 1)^2))^k. - Antonio Graciá Llorente, Jun 13 2024
Equals Integral_{x=0..2} sqrt(8 - x^2) dx - 2 (see Ambrisi and Rizzi). - Stefano Spezia, Jul 21 2024
Equals 3 + 4*Sum_{k>0} (-1)^(k+1)/(4*k*(1+k)*(1+2*k)) (see Wells at p. 53). - Stefano Spezia, Aug 31 2024
Equals 4*Integral_{x=0..1} sqrt(1 - x^2) dx = lim_{n->oo} (4/n^2)*Sum_{k=0..n} sqrt(n^2 - k^2) (see Finch). - Stefano Spezia, Oct 19 2024
Equals Beta(1/2,1/2) (see Shamos). - Stefano Spezia, Jun 03 2025
From Kritsada Moomuang, Jun 18 2025: (Start)
Equals 2 + Integral_{x=0..1} 1/(sqrt(x)*(1 + sqrt(1 - x))) dx.
Equals 2 + Integral_{x=0..1} log(1 + sqrt(1 - x))/sqrt(x) dx. (End)
Pi = 2*arccos(1/phi) + arccos(1/phi^3) = 4*arcsin(1/phi) + 2*arcsin(1/phi^3) where phi = (1+sqrt(5))/2. - Chittaranjan Pardeshi, Jul 02 2025
Pi = Sum_{n >= 0} zeta(2*n)*(2^(2*n - 1) - 1)/2^(4*n - 3). - Andrea Pinos, Jul 29 2025

Additional comments from William Rex Marshall, Apr 20 2001

A000035 Period 2: repeat [0, 1]; a(n) = n mod 2; parity of n.

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0

Offset: 0

N. J. A. Sloane

Least significant bit of n, lsb(n).
Also decimal expansion of 1/99.
Also the binary expansion of 1/3. - Robert G. Wilson v, Sep 01 2015
a(n) = A134451(n) mod 2. - Reinhard Zumkeller, Oct 27 2007 [Corrected by Jianing Song, Nov 22 2019]
Characteristic function of odd numbers: a(A005408(n)) = 1, a(A005843(n)) = 0. - Reinhard Zumkeller, Sep 29 2008
A102370(n) modulo 2. - Philippe Deléham, Apr 04 2009
Base b expansion of 1/(b^2-1) for any b >= 2 is 0.0101... (A005563 has b^2-1). - Rick L. Shepherd, Sep 27 2009
Let A be the Hessenberg n X n matrix defined by: A[1,j] = j mod 2, A[i,i] := 1, A[i,i-1] = -1, and A[i,j] = 0 otherwise. Then, for n >= 1, a(n) = (-1)^n*charpoly(A,1). - Milan Janjic, Jan 24 2010
From R. J. Mathar, Jul 15 2010: (Start)
The sequence is the principal Dirichlet character of the reduced residue system mod 2 or mod 4 or mod 8 or mod 16 ...
Associated Dirichlet L-functions are for example L(2,chi) = Sum_{n>=1} a(n)/n^2 == A111003,
or L(3,chi) = Sum_{n>=1} a(n)/n^3 = 1.05179979... = 7*A002117/8,
or L(4,chi) = Sum_{n>=1} a(n)/n^4 = 1.014678... = A092425/96. (End)
Also parity of the nonnegative integers A001477. - Omar E. Pol, Jan 17 2012
a(n) = (4/n), where (k/n) is the Kronecker symbol. See the Eric Weisstein link. - Wolfdieter Lang, May 28 2013
Also the inverse binomial transform of A131577. - Paul Curtz, Nov 16 2016 [an observation forwarded by Jean-François Alcover]
The emanation sequence for the globe category. That is take the globe category, take the corresponding polynomial comonad, consider its carrier polynomial as a generating function, and take the corresponding sequence. - David Spivak, Sep 25 2020
For n > 0, a(n) is the alternating sum of the product of n increasing and n decreasing odd factors. For example, a(4) = 1*7 - 3*5 + 5*3 - 7*1 and a(5) = 1*9 - 3*7 + 5*5 - 7*3 + 9*1. - Charlie Marion, Mar 24 2022

			G.f. = x + x^3 + x^5 + x^7 + x^9 + x^11 + x^13 + x^15 + ... - _Michael Somos_, Feb 20 2024

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

David Wasserman, Table of n, a(n) for n = 0..1000
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
F. Javier de Vega, An extension of Furstenberg's theorem of the infinitude of primes, arXiv:2003.13378 [math.NT], 2020.
Clark Kimberling, A Combinatorial Classification of Triangle Centers on the Line at Infinity, J. Int. Seq., Vol. 22 (2019), Article 19.5.4.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Eric Weisstein's World of Mathematics, Dirichlet Series Generating Function
Eric Weisstein's World of Mathematics, Kronecker Symbol
A. K. Whitford, Binet's Formula Generalized, Fibonacci Quarterly, Vol. 15, No. 1, 1979, pp. 21, 24, 29
Index entries for "core" sequences
Index entries for sequences that are fixed points of mappings
Index entries for characteristic functions
Index entries for linear recurrences with constant coefficients, signature (0,1).

Ones complement of A059841.
Cf. A053644 for most significant bit.
This is Guy Steele's sequence GS(1, 2) (see A135416).
Period k zigzag sequences: this sequence (k=2), A007877 (k=4), A260686 (k=6), A266313 (k=8), A271751 (k=10), A271832 (k=12), A279313 (k=14), A279319 (k=16), A158289 (k=18).
Cf. A154955 (Mobius transform), A131577 (binomial transform).
Cf. A111003 (Dgf at s=2), A233091 (Dgf at s=3), A300707 (Dgf at s=4).
Parity of A005811.

a000035 n = n `mod` 2  -- James Spahlinger, Oct 08 2012

a000035_list = cycle [0,1]  -- Reinhard Zumkeller, Jan 06 2012

A000035 := n->n mod 2;
[ seq(i mod 2, i=0..100) ];

PadLeft[{},110,{0,1}] (* Harvey P. Dale, Sep 25 2011 *)

A000035(n):=mod(n,2)$
makelist(A000035(n),n,0,30); /* Martin Ettl, Nov 12 2012 */

PARI
```
a(n)=n%2;
```

a(n)=direuler(p=1,100,if(p==2,1,1/(1-X)))[n] /* Ralf Stephan, Mar 27 2015 */

def A000035(n): return n & 1 # Chai Wah Wu, May 25 2022

(define (A000035 n) (mod n 2)) ;; For R6RS. Use modulo in older Schemes like MIT/GNU Scheme. - Antti Karttunen, Mar 21 2017

a(n) = (1 - (-1)^n)/2.
a(n) = n mod 2.
a(n) = 1 - a(n-1).
Multiplicative with a(p^e) = p mod 2. - David W. Wilson, Aug 01 2001
G.f.: x/(1-x^2). E.g.f.: sinh(x). - Paul Barry, Mar 11 2003
a(n) = (A000051(n) - A014551(n))/2. - Mario Catalani (mario.catalani(AT)unito.it), Aug 30 2003
a(n) = ceiling((-2)^(-n-1)). - Reinhard Zumkeller, Apr 19 2005
Dirichlet g.f.: (1-1/2^s)*zeta(s). - R. J. Mathar, Mar 04 2011
a(n) = ceiling(n/2) - floor(n/2). - Arkadiusz Wesolowski, Sep 16 2012
a(n) = ceiling( cos(Pi*(n-1))/2 ). - Wesley Ivan Hurt, Jun 16 2013
a(n) = floor((n-1)/2) - floor((n-2)/2). - Mikael Aaltonen, Feb 26 2015
Dirichlet g.f.: L(chi(2),s) with chi(2) the principal Dirichlet character modulo 2. - Ralf Stephan, Mar 27 2015
a(n) = 0^^n = 0^(0^(0...)) (n times), where we take 0^0 to be 1. - Natan Arie Consigli, May 02 2015
Euler transform and inverse Moebius transform of length 2 sequence [0, 1]. - Michael Somos, Feb 20 2024

A013662 Decimal expansion of zeta(4).

Original entry on oeis.org

1, 0, 8, 2, 3, 2, 3, 2, 3, 3, 7, 1, 1, 1, 3, 8, 1, 9, 1, 5, 1, 6, 0, 0, 3, 6, 9, 6, 5, 4, 1, 1, 6, 7, 9, 0, 2, 7, 7, 4, 7, 5, 0, 9, 5, 1, 9, 1, 8, 7, 2, 6, 9, 0, 7, 6, 8, 2, 9, 7, 6, 2, 1, 5, 4, 4, 4, 1, 2, 0, 6, 1, 6, 1, 8, 6, 9, 6, 8, 8, 4, 6, 5, 5, 6, 9, 0, 9, 6, 3, 5, 9, 4, 1, 6, 9, 9, 9, 1

Offset: 1

N. J. A. Sloane

			1.082323233711138191516003696541167...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 811.
Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 89, Exercise.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 262.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section F17, Series associated with the zeta-function, p. 391.
L. D. Landau and E. M. Lifschitz, Band V, Statistische Physik, Akademie Verlag, 1966, pp. 172 and 180-181.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 162.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, p. 33.

Harry J. Smith, Table of n, a(n) for n = 1..20000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Peter Bala, New series for old functions.
D. H. Bailey, J. M. Borwein, and D. M. Bradley, Experimental determination of Apéry-like identities for zeta(4n+2), arXiv:math/0505270 [math.NT], 2005-2006.
D. Borwein and J. M. Borwein, On an intriguing integral and some series related to zeta(4) Proc. Amer. Math. Soc., Vol. 123, No.4, April 1995.
J. M. Borwein, D. J. Broadhurst, and J. Kamnitzer, Central binomial sums, multiple Clausen values and zeta values arXiv:hep-th/0004153, 2000.
Leonhard Euler, On the sums of series of reciprocals, arXiv:math/0506415 [math.HO], 2005-2008.
Leonhard Euler, De summis serierum reciprocarum, E41.
Raffaele Marcovecchio and Wadim Zudilin, Hypergeometric rational approximations to zeta(4), arXiv:1905.12579 [math.NT], 2019.
Romeo Meštrović, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv:1202.3670 [math.HO], 2012. - From _N. J. A. Sloane_, Jun 13 2012
Jean-Christophe Pain, An integral representation for zeta(4), arXiv:2309.00539 [math.NT], 2023.
Michael Penn, Finding a closed form for zeta(4), YouTube video, 2022.
Simon Plouffe, Pi^4/90 to 100000 digits.
Simon Plouffe, Zeta(4) or Pi^4/90 to 10000 places.
Simon Plouffe, Zeta(2) to Zeta(4096) to 2048 digits each (gzipped file).
K. Ramachandra, On series integrals and continued fractions I, Hardy-Ramanujan Journal, Vol. 4 (1981), pp. 1-11.
Carsten Schneider and Wadim Zudilin, A case study for zeta(4), arXiv:2004.08158 [math.NT], 2020.
Chuanan Wei, Some fast convergent series for the mathematical constants zeta(4) and zeta(5), arXiv:2303.07887 [math.CO], 2023.
Index entries for transcendental numbers.

Cf. A013661, A002117, A013663, A013664, A013665, A013666, A013667, A013668, A013669, A013670, A013671, A013672, A013673, A013674, A013675, A013676, A013677, A013678, A293904.
Cf. A231535, A337711.
Cf. A001008, A002805.
See also the extensive crossref table in A308637.

SetDefaultRealField(RealField(110)); L:=RiemannZeta(); Evaluate(L,4); // G. C. Greubel, May 30 2019

evalf(Pi^4/90,120); # Muniru A Asiru, Sep 19 2018

RealDigits[Zeta[4],10,120][[1]] (* Harvey P. Dale, Dec 18 2012 *)

ev(zeta(4),numer) ; /* R. J. Mathar, Feb 27 2012 */

default(realprecision, 20080); x=Pi^4/90; for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b013662.txt", n, " ", d)); \\ Harry J. Smith, Apr 29 2009

numerical_approx(zeta(4), digits=100) # G. C. Greubel, May 30 2019

zeta(4) = Pi^4/90 = A092425/90. - Harry J. Smith, Apr 29 2009
From Peter Bala, Dec 03 2013: (Start)
Definition: zeta(4) := Sum_{n >= 1} 1/n^4.
zeta(4) = (4/17)*Sum_{n >= 1} ( (1 + 1/2 + ... + 1/n)/n )^2 and
zeta(4) = (16/45)*Sum_{n >= 1} ( (1 + 1/3 + ... + 1/(2*n-1))/n )^2 (see Borwein and Borwein).
zeta(4) = (256/90)*Sum_{n >= 1} n^2*(4*n^2 + 3)*(12*n^2 + 1)/(4*n^2 - 1)^5.
Series acceleration formulas:
zeta(4) = (36/17)*Sum_{n >= 1} 1/( n^4*binomial(2*n,n) ) (Comtet)
        = (36/17)*Sum_{n >= 1} P(n)/( (2*n*(2*n - 1))^4*binomial(4*n,2*n) )
        = (36/17)*Sum_{n >= 1} Q(n)/( (3*n*(3*n - 1)*(3*n - 2))^4*binomial(6*n,3*n) ),
where P(n) = 80*n^4 - 48*n^3 + 24*n^2 - 8*n + 1 and Q(n) = 137781*n^8 - 275562*n^7 + 240570*n^6 - 122472*n^5 + 41877*n^4 - 10908*n^3 + 2232*n^2 - 288*n + 16 (see section 8 in the Bala link). (End)
zeta(4) = 2/3*2^4/(2^4 - 1)*( Sum_{n even} n^2*p(n)/(n^2 - 1)^5 ), where p(n) = 3*n^4 + 10*n^2 + 3 is a row polynomial of A091043. See A013664, A013666, A013668 and A013670. - Peter Bala, Dec 05 2013
zeta(4) = Sum_{n >= 1} ((floor(sqrt(n))-floor(sqrt(n-1)))/n^2). - Mikael Aaltonen, Jan 18 2015
zeta(4) = Product_{k>=1} 1/(1 - 1/prime(k)^4). - Vaclav Kotesovec, May 02 2020
From Wolfdieter Lang, Sep 16 2020: (Start)
zeta(4) = (1/3!)*Integral_{x=0..oo} x^3/(exp(x) - 1) dx. See Abramowitz-Stegun, 23.2.7., for s=2, p. 807, and Landau-Lifschitz, Band V, p. 172, eq. (4), for x=4. See also A231535.
zeta(4) = (4/21)*Integral_{x=0..oo} x^3/(exp(x) + 1) dx. See Abramowitz-Stegun, 23.2.8., for s=2, p. 807, and Landau-Lifschitz, Band V, p. 172, eq. (1), for x=4. See also A337711. (End)
zeta(4) = (72/17) * Integral_{x=0..Pi/3} x*(log(2*sin(x/2)))^2. See Richard K. Guy reference. - Bernard Schott, Jul 20 2022
From Peter Bala, Nov 12 2023: (Start)
zeta(4) = 1 + (4/3)*Sum_{k >= 1} (1 - 2*(-1)^k)/(k*(k + 1)^4*(k + 2)) = 35053/32400 + 48*Sum_{k >= 1} (1 - 2*(-1)^k)/(k*(k + 1)*(k + 2)*(k + 3)^4*(k + 4)*(k + 5)*(k + 6)).
More generally, it appears that for n >= 0, zeta(4) = c(n) + (4/3)*(2*n + 1)!^2 * Sum_{k >= 1} (1 - 2*(-1)^k)/( (k + 2*n + 1)^3*Product_{i = 0..4*n+2} (k + i) ), where {c(n) : n >= 0} is a sequence of rational approximations to zeta(4) beginning [1, 35053/32400, 2061943067/ 1905120000, 18594731931460103/ 17180389306080000, 257946156103293544441/ 238326360453941760000, ...]. (End)
From Peter Bala, Apr 27 2025: (Start)
zeta(4) = 1/4! * Integral_{x >= 0} x^4 * exp(x)/(exp(x) - 1)^2 dx = 8/7 * 1/4! * Integral_{x >= 0} x^4 * exp(x)/(exp(x) + 1)^2 dx.
zeta(4) = 1/5! * Integral_{x >= 0} x^5 * exp(x)*(exp(x) + 1)/(exp(x) - 1)^3 dx = 1/(3*5*7) * Integral_{x >= 0} x^5 * exp(x)*(exp(x) - 1)/(exp(x) + 1)^3 dx. (End)
10*zeta(4) = Sum_{k>=1} H(k)^3/(k*(k+1)), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number (Ramachandra, 1981). - Amiram Eldar, May 30 2025
zeta(4) = Integral_{x=0..1} Li(3,x)/x dx, where Li(n,x) is the polylogarithm function. - Kritsada Moomuang, Jun 14 2025
zeta(4) = Sum_{i, j >= 1} 1/(i^3*j*binomial(i+j, i)) = 4/3 * Sum_{i, j >= 1} 1/(i^2*j^2*binomial(i+j, i)). - Peter Bala, Aug 03 2025

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A000796 Decimal expansion of Pi (or digits of Pi).

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A000035 Period 2: repeat [0, 1]; a(n) = n mod 2; parity of n.

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A013662 Decimal expansion of zeta(4).

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A152649 Decimal expansion of Pi^4/72.

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A197110 Decimal expansion of Pi^4/120.

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