OFFSET
0,1
COMMENTS
Equals Integral_{x>=0} sin(4*x)/(4*x) dx. - Jean-François Alcover, Feb 28 2013
Consider 4 circles inscribed in a square. Inscribe a square in each circle. And finally, inscribe 4 circles inside each four small squares. Totally we get 16 small circles. Pi/8 is the ratio of the area of the 16 small circles to the area of initial square. See the link. - Kirill Ustyantsev, Apr 30 2020
Volumetric flux, normalized by R^4 |dp/dx|/mu, of a slowly-moving fluid of dynamic viscosity mu in a pipe of radius R driven by pressure gradient dp/dx. Batchelor (1967) discusses the conditions required for this flow, called Poiseuille flow. - Chris R. Rehmann, Dec 28 2025
REFERENCES
G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, 1967, section 4.2., eq. 4.2.6.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.4, p. 492.
Stephen Hawking, God Created the Integers: The Mathematical Breakthroughs That Changed History, Running Press, 2007. See p. 576.
LINKS
Ivan Panchenko, Table of n, a(n) for n = 0..1000
H. F. Sandham, Problem 4191, The American Mathematical Monthly, Vol. 53, No. 2 (1946), p. 103; Summation, Solution to Problem 4191 by H. E. Fettis, ibid., Vol. 54, No. 6 (1947), pp. 347-349.
Kirill Ustyantsev, Geometric sense of Pi/8.
FORMULA
From Peter Bala, Nov 15 2016: (Start)
Pi/8 = Sum_{k >= 1} (-1)^k/((2*k - 3)*(2*k - 1)*(2*k + 1)).
More generally, for n >= 0 we have 1/(2*n)! * Pi/4 = Sum_{k >= 1} (-1)^(k+n-1) * 1/Product_{j = -n..n} (2*k + 2*j - 1): when n = 0 we get the Madhava-Gregory-Leibniz series for Pi/4.
For N divisible by 4, we have the asymptotic expansion Pi/8 - Sum_{k = 1..N/2} (-1)^k/((2*k - 3)*(2*k - 1)*(2*k + 1)) ~ -1/2*(1/N^3 - 2/N^5 + 31/N^7 - 692/N^9 + ...), where the sequence of unsigned coefficients [1, 2, 31, 692, ...] equals A024235. (End)
Equals Integral_{x = 0..1} x*sqrt(1 - x^4) dx. - Peter Bala, Oct 27 2019
Equals Integral_{x = 0..oo} sin(x)^6/x^4 dx = Sum_{n >= 1} sin(n)^6/n^4, by the Abel-Plana formula. - Peter Bala, Nov 04 2019
From Amiram Eldar, Jul 12 2020: (Start)
Equals arctan(sqrt(2) - 1).
Equals Sum_{k>=0} (-1)^k/(4*k+2).
Equals Sum_{k>=0} 1/((4*k+1)*(4*k+3)) = Sum_{k>=0} 1/A001539(k).
Equals Integral_{x=0..oo} dx/(x^2 + 16).
Equals Integral_{x=0..oo} dx/(x^4 + 4) = Integral_{x=0..oo} x/(x^4 + 4) dx.
Equals Integral_{x=0..oo} x/(x^4 + 1)^2 dx = Integral_{x=0..1} x/(x^4 + 1) dx.
Equals Integral_{x=0..1} x * arcsin(x) dx. (End)
From Kritsada Moomuang, Jun 18 2025: (Start)
Equals Integral_{x=0..oo} (x*log(x + 1))/((x^2 + 1)^2) dx.
Equals Integral_{x=0..oo} (x^3 - 3*x + 3*arctan(x))/(3*x^5) dx. (End)
Equals Sum_{k>=1} (-1)^(k+1)/((2*k-1)*cosh((2*k-1)*Pi/2)) (Sandham, 1946). - Amiram Eldar, Apr 08 2026
Equals Integral_{x=0..oo} 1 - Sum_{t>=0} (1 - 12*(t*x)^2)/(1 + 4*(t*x)^2)^3 dx. - Mats Granvik, May 23 2026
EXAMPLE
Pi/8 = 0.392699081698724154807830422909937860524646174921888227621868... - Vladimir Joseph Stephan Orlovsky, Dec 02 2009
MATHEMATICA
RealDigits[N[Pi/8, 6! ]] (* Vladimir Joseph Stephan Orlovsky, Dec 02 2009 *)
PROG
(PARI)
default(realprecision, 1002);
eval(vecextract(Vec(Str(sumalt(n=0, (-1)^(n)/(4*n+2)))), "3..-2")) \\ Gheorghe Coserea, Oct 06 2015
(Magma) pi:=Pi(RealField(110)); Reverse(Intseq(Floor(10^100*(pi)/8))); // Vincenzo Librandi, Oct 07 2015
CROSSREFS
KEYWORD
AUTHOR
STATUS
approved