A063448 Decimal expansion of Pi * sqrt(2).

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

Offset

1,1

Comments

Hypotenuse of the right triangle with legs Pi and Pi. - Zak Seidov, May 04 2005

Circumference of the circumcircle of the unit square. - Jonathan Sondow, Nov 23 2017

Half-perimeter of the closed curve with implicit Cartesian equation x^2 + y^2 = abs(x) + abs(y). - Stefano Spezia, Oct 20 2020

Links

Harry J. Smith, Table of n, a(n) for n = 1..20000

Peter Bala, Series for sqrt(2)*Pi

Eric Weisstein's World of Mathematics, Gauss Multiplication Formula.

Wikipedia, Bailey-Borwein-Plouffe formula.

Index entries for transcendental numbers

Formula

Equals Gamma(1/4)*Gamma(3/4). - Jean-François Alcover, Nov 24 2014

From Amiram Eldar, Aug 06 2020: (Start)

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

Equals Integral_{x=0..oo} log(1 + 2/x^2) dx.

Equals Integral_{x=-oo..oo} exp(x/4)/(exp(x) + 1) dx.

Equals Integral_{x=0..2*Pi} 1/(cos(x)^2 + 1) dx = Integral_{x=0..2*Pi} 1/(sin(x)^2 + 1) dx. (End)

From Andrea Pinos, Jul 03 2023: (Start)

Equals (Product_{k=1..4} Gamma(k/8)*Gamma(1 - k/8))^(1/4).

General result: 2*Pi/(4*y)^(1/(2*y)) = (Product_{k=1..y} Gamma(k/(2*y))*Gamma(1 - k/(2*y)) )^(1/y). (End)

From Peter Bala, Oct 22 2023: (Start)

sqrt(2)*Pi = 4 + 8*Sum_{n >= 0} (-1)^n/(16*n^2 + 32*n + 15). See A141759.

In the following the Eisenstein summation convention is assumed: that is,

Sum_{n = -oo..oo} means Limit_{N -> oo} Sum_{n = -N..N}:

sqrt(2)*Pi = 4*Sum_{n = -oo..oo} (-1)^n/(4*n + 1).

More generally, it appears that for k >= 0, k not of the form 4*m + 1,

sqrt(2)*Pi = -sign(cos(Pi*(k - 3)/4)) * 4*(2^floor(k/2))*k! * Sum_{n = -oo..oo} (-1)^n/((4*n + 1)*(4*n + 3)*...*(4*n + 2*k + 1)) (verified up to k = 50).

sqrt(2)*Pi = (2^4)*Sum_{n >= 0} (-1)^n * (2*n + 1)/((4*n + 1)*(4*n + 3)) = 512/105 - (2^6)*4!*Sum_{n >= 0} (-1)^n * (2*n + 3)/((4*n + 1)*(4*n + 3)*...*(4*n + 11)).

sqrt(2)*Pi = 4 + (2^3)*Sum_{n >= 0} (-1)^n * (4*n + 1)/((4*n + 1)*(4*n + 3)*(4*n + 5)) = 1408/315 - (2^5)*5!*Sum_{n >= 0} (-1)^n * (4*n + 1)/((4*n + 1)*(4*n + 3)*...*(4*n + 13)).

sqrt(2)*Pi = 16/3 - (2^4)*3!*Sum_{n >= 0} (-1)^n/((4*n + 1)*(4*n + 3)*(4*n + 5)*(4*n + 7)) = 14848/3465 + (2^6)*7!*Sum_{n >= 0} (-1)^n/((4*n + 1)*(4*n + 3)*...*(4*n + 15)). (End)

From Peter Bala, Nov 19 2023: (Start)

sqrt(2)*Pi = 512*Sum_{k >= 1} (-1)^(k+1) * k^2/((16*k^2 - 1)*(16*k^2 - 9)).

This is the case n = 1 of the more general result: for n >= 1,

sqrt(2)*Pi = (-1)^(n+1) * 2^(n+7) * (2*n)!/(2*n - 1) * Sum_{k >= 1} (-1)^(k+1) * k^2/( Product_{i = 0..n} (16*k^2 - (2*i+1)^2) ). Cf. A334422. (End)

Example

4.4428829381583662470158809900606936986146216893756902230853...

Mathematica

RealDigits[N[Pi*Sqrt[2], 200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Mar 21 2011*)

Prog

(PARI) \p 400; Pi * sqrt(2)
(PARI) default(realprecision, 20080); x=Pi*sqrt(2); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b063448.txt", n, " ", d)) \\ Harry J. Smith, Aug 21 2009
(Python)  # Use some guard digits when computing.
# BBP formula (1/8) P(1, 64, 12, (32, 0, 8, 0, 8, 0, -4, 0, -1, 0, -1, 0))
from decimal import Decimal as dec, getcontext
def BBPpisqrt2(n: int) -> dec:
    getcontext().prec = n
    s = dec(0); f = dec(1); g = dec(64)
    for k in range(int(n * 0.5536546824812272) + 1):
        twk = dec(12 * k)
        s += f * ( dec(32) / (twk + 1) + dec(8)  / (twk + 3)
                 + dec(8)  / (twk + 5) - dec(4)  / (twk + 7)
                 - dec(1)  / (twk + 9) - dec(1)  / (twk + 11))
        f /= g
    return s / dec(8)
print(BBPpisqrt2(200))  # Peter Luschny, Nov 03 2023

Crossrefs

Cf. A063447 (continued fraction), A093954, A153799, A193887, A244976, A247719.

Sequence in context: A182565 A016708 A105724 * A232526 A358328 A117499

Adjacent sequences: A063445 A063446 A063447 * A063449 A063450 A063451

Keyword

cons,nonn

Author

Jason Earls, Jul 24 2001

Extensions

Edited by N. J. A. Sloane, May 05 2007

Corrected by Neven Juric, Apr 24 2008

Status

approved