A122952 -id:A122952 - cp's OEIS frontend

A019669 Decimal expansion of Pi/2.

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

Offset: 1

N. J. A. Sloane

With offset 2, decimal expansion of 5*Pi. - Omar E. Pol, Oct 03 2013
Decimal expansion of the number of radians in a quadrant. - John W. Nicholson, Oct 07 2013
Not the same as A085679. First differing term occurs at 10^-49, as list -49, or 51st counting term (a(-49)= 5 and A085679(-49) = 4). - John W. Nicholson, Oct 07 2013
5*Pi is also the surface area of a sphere whose diameter equals the square root of 5. More generally x*Pi is also the surface area of a sphere whose diameter equals the square root of x. - Omar E. Pol, Dec 22 2013
Pi/2 is also the radius of a sphere whose surface area equals the volume of the circumscribed cube. - Omar E. Pol, Dec 27 2013

			Pi/2 = 1.570796326794896619231321691639751442098584699...
5*Pi = 15.70796326794896619231321691639751442098584699...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 1.4.1 and 1.4.2, pp. 20-21.

Harry J. Smith, Table of n, a(n) for n = 1..20000
David H. Bailey and Richard E. Crandall, On the Random Character of Fundamental Constant Expansions, Experimental Mathematics, Vol. 10 (2001), Issue 2, p. 185 (preprint draft).
Richard J. Mathar, Chebyshev approximation of x^m (-log x)^l in the interval 0 <= x <= 1, arXiv:2408.15212 [math.CA], 2024. See p. 2.
Michael Penn, A nice sum from the Harvard MIT math trust, YouTube video, 2022.
L. D. Servi, Nested Square Roots of 2, The American Mathematical Monthly 110:4 (Apr. 2003), pp. 326-330.
Johan Wästlund, An Elementary Proof of the Wallis Product Formula for pi, The American Mathematical Monthly 114:10 (Dec. 2007), pp. 914-917.
Eric W. Weisstein and Jonathan Sondow, Wallis Formula, MathWorld.
Wikipedia, Viète's formula.
Index entries for transcendental numbers.

Cf. A053300 (continued fraction), A060294 (2/Pi).

Cf. A000796, A019692, A122952, A019694 (Pi through 4*Pi), A106854.

Digits:=100: evalf(Pi/2); # Wesley Ivan Hurt, Oct 26 2016

RealDigits[N[Pi/2, 200]] (* Vladimir Joseph Stephan Orlovsky, Dec 02 2009 *)

default(realprecision, 20080); x=Pi/2; for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b019669.txt", n, " ", d)); \\ Harry J. Smith, May 31 2009

Pi/2 = log(i)/i, where i = sqrt(-1). - Eric Desbiaux, Jun 27 2009
Pi/2 = Product_{n>=1} (n/(n+1))^((-1)^n) = 2 * 2/3 * 4/3 * 4/5 * 6/5 * 6/7 * 8/7 * 8/9 * 10/9 * ... (Wallis formula). - William Keith and Alonso del Arte, Jun 24 2012
Equals Sum_{k>1} 2^k/binomial(2*k,k). - Bruno Berselli, Sep 11 2015
The previous result is the particular case n = 1 of the more general identity: Pi/2 = 4^(n-1) * n!/(2*n)! * Sum_{k >= 2} 2^(k+1)*(k + n - 1)!*(k + 2*n - 2)!/(2*k + 2*n - 2)! valid for n = 0,1,2,... . - Peter Bala, Oct 26 2016
Pi/2 = Product_{n>=1} (4*n^2)/(4*n^2-1). - Fred Daniel Kline, Oct 29 2016
Pi/2 = lim_{n->oo} F(2^(n+3))/2, with one half of the area of a regular 2^(n+3)-gon, for n >= 0, inscribed in the unit circle, written as iterated square roots of 2 as F(2^(n+3))/2 = 2^n*sqrt(2 + sq2(n)), with sq2(n) = sqrt(2 + sq2(n-1)), n >= 1, with input sq2(0) = 0 (2 appears n times in sq2(n)). Viète's infinite product formula works with the partial product F(2^(n+2))/2 = Product_{j=1..n} (2/sq2(j)), n >= 1, which corresponds to the above given formula. - Wolfdieter Lang, Jul 06 2018
Pi/2 = Integral_{x = 0..oo} sin(x)^2/x^2 dx = 1/2 + Sum_{n >= 1} sin(n)^2/n^2, by the Abel-Plana formula. - Peter Bala, Nov 05 2019
From Amiram Eldar, Aug 15 2020: (Start)
Equals Sum_{k>=0} k!/(2*k + 1)!!.
Equals Sum_{k>=0} (-1)^k/(k + 1/2).
Equals Integral_{x=0..oo} 1/(x^2 + 1) dx.
Equals Integral_{x=0..oo} sin(x)/x dx.
Equals Integral_{x=0..oo} exp(x/2)/(exp(x) + 1) dx.
Equals Product_{p prime > 2} p/(p + (-1)^((p-1)/2)). (End)
Pi/2 = Integral_{x = 0..oo} 1/(1 - x^2 + x^4) dx = (1 + 2/3 + 1/5) - (1/7 + 2/9 + 1/11) + (1/13 + 2/15 + 1/17) - .... - Peter Bala, Jul 22 2022
Equals arcsin(9/10) + sqrt(19)*Sum_{k >= 1} A106854(k-1)/(k*10^k) (see Bailey and Crandall, 2001). - Paolo Xausa, Jul 15 2024
Equals 2F1(1/2,1/2 ; 3/2; 1). - R. J. Mathar, Aug 20 2024
Pi/2 = [1;1,1/2,1/3,...,1/n,...] by Wallis's approximation. - Thomas Ordowski, Oct 19 2024
From Stefano Spezia, Oct 21 2024: (Start)
Equals Sum_{k>=0} 2^k/((2*k + 1)*binomial(2*k,k)) (see Finch).
Equals Limit_{n->oo} 2^(4*n)/((2*n + 1)*binomial(2*n,n)^2) (see Finch). (End)
Equals Integral_{x=-oo..oo} sech((2*x^3 + x^2 - 5*x)/(x^2 - 1)) dx. - Kritsada Moomuang, May 29 2025

A019670 Decimal expansion of Pi/3.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Dec 11 1996

With an offset of zero, also the decimal expansion of Pi/30 ~ 0.104719... which is the average arithmetic area  of the 0-winding sectors enclosed by a closed Brownian planar path, of a given length t, according to Desbois, p. 1. - Jonathan Vos Post, Jan 23 2011
Polar angle (or apex angle) of the cone that subtends exactly one quarter of the full solid angle. See comments in A238238. - Stanislav Sykora, Jun 07 2014
60 degrees in radians. - M. F. Hasler, Jul 08 2016
Volume of a quarter sphere of radius 1. - Omar E. Pol, Aug 17 2019
Also smallest positive zero of Sum_{k>=1} cos(k*x)/k = -log(2*|sin(x/2)|). Proof of this identity: Sum_{k>=1} cos(k*x)/k = Re(Sum_{k>=1} exp(k*x*i)/k) = Re(-log(1-exp(x*i))) = -log(2*|sin(x/2)|), x != 2*m*Pi, where i = sqrt(-1). - Jianing Song, Nov 09 2019
The area of a circle circumscribing a unit-area regular dodecagon. - Amiram Eldar, Nov 05 2020

			Pi/3 = 1.04719755119659774615421446109316762806572313312503527365831486...
From _Peter Bala_, Nov 16 2016: (Start)
Case n = 1. Pi/3 = 18 * Sum_{k >= 0} (-1)^(k+1)( 1/((6*k - 5)*(6*k + 1)*(6*k + 7)) + 1/((6*k - 1)*(6*k + 5)*(6*k + 11)) ).
Using the methods of Borwein et al. we can find the following asymptotic expansion for the tails of this series: for N divisible by 6 there holds Sum_{k >= N/6} (-1)^(k+1)( 1/((6*k - 5)*(6*k + 1)*(6*k + 7)) + 1/((6*k - 1)*(6*k + 5)*(6*k + 11)) ) ~ 1/N^3 + 6/N^5 + 1671/N ^7 - 241604/N^9 + ..., where the sequence [1, 0, 6, 0, 1671, 0, -241604, 0, ...] is the sequence of coefficients in the expansion of ((1/18)*cosh(2*x)/cosh(3*x)) * sinh(3*x)^2 = x^2/2! + 6*x^4/4! + 1671*x^6/6! - 241604*x^8/8! + .... Cf. A024235, A278080 and A278195. (End)

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.3, p. 489.

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Kunle Adegoke, Infinite arctangent sums involving Fibonacci and Lucas numbers, arXiv:1603.08097 [math.NT], 2016.
J. M. Borwein, P. B. Borwein, and K. Dilcher, Pi, Euler numbers and asymptotic expansions, Amer. Math. Monthly, 96 (1989), 681-687.
Jean Desbois and Stéphane Ouvry, Algebraic and arithmetic area for m planar Brownian paths, arXiv:1101.4135 [math-ph], Jan 21, 2011.
Index entries for transcendental numbers.

Cf. A000796 (Pi), A019669 (Pi/2), A019692 (2*Pi), A122952 (3*Pi), A003881(Pi/4), A137914, A238238, A002388, A091925, A093954, A016910, A019694, A136017, A352324.

Integral_{x=0..oo} 1/(1+x^m) dx: A013661 (m=2), A248897 (m=3), A093954 (m=4), A352324 (m=5), this sequence (m=6), A352125 (m=8), A094888 (m=10).

RealDigits[N[Pi/3,6! ]] (* Vladimir Joseph Stephan Orlovsky, Dec 02 2009 *)

Pi/3 \\ Charles R Greathouse IV, Sep 08 2011

N=150; default(realprecision, 3+N); digits(Pi*10^N\3) \\ M. F. Hasler, Jul 08 2016

A third of A000796, a sixth of A019692, the square root of A100044.
Sum_{k >= 0} (-1)^k/(6k+1) + (-1)^k/(6k+5). - Charles R Greathouse IV, Sep 08 2011
Product_{k >= 1}(1-(6k)^(-2))^(-1). - Fred Daniel Kline, May 30 2013
From Peter Bala, Feb 05 2015: (Start)
Pi/3 = Sum {k >= 0} binomial(2*k,k)*1/(2*k + 1)*(1/16)^k = 2F1(1/2,1/2;3/2;1/4). Similar series expansions hold for Pi^2 (A002388), Pi^3 (A091925) and Pi/(2*sqrt(2)) (A093954.)
The integer sequences A(n) := 4^n*(2*n + 1)! and B(n) := A(n)*( Sum {k = 0..n} binomial(2*k,k)*1/(2*k + 1)*(1/16)^k ) both satisfy the second-order recurrence equation u(n) = (20*n^2 + 4*n + 1)*u(n-1) - 8*(n - 1)*(2*n - 1)^3*u(n-2). From this observation we can obtain the continued fraction expansion Pi/3 = 1 + 1/(24 - 8*3^3/(89 - 8*2*5^3/(193 - 8*3*7^3/(337 - ... - 8*(n - 1)*(2*n - 1)^3/((20*n^2 + 4*n + 1) - ... ))))). Cf. A002388 and A093954. (End)
Equals Sum_{k >= 1} arctan(sqrt(3)*L(2k)/L(4k)) where L=A000032. See also A005248 and A056854. - Michel Marcus, Mar 29 2016
Equals Product_{n >= 1} A016910(n) / A136017(n). - Fred Daniel Kline, Jun 09 2016
Equals Integral_{x=-oo..oo} sech(x)/3 dx. - Ilya Gutkovskiy, Jun 09 2016
From Peter Bala, Nov 16 2016: (Start)
Euler's series transformation applied to the series representation Pi/3 = Sum_{k >= 0} (-1)^k/(6*k + 1) + (-1)^k/(6*k + 5) given above by Greathouse produces the faster converging series Pi/3 = (1/2) * Sum_{n >= 0} 3^n*n!*( 1/(Product_{k = 0..n} (6*k + 1)) + 1/(Product_{k = 0..n} (6*k + 5)) ).
The series given above by Greathouse is the case n = 0 of the more general result Pi/3 = 9^n*(2*n)! * Sum_{k >= 0} (-1)^(k+n)*( 1/(Product_{j = -n..n} (6*k + 1 + 6*j)) + 1/(Product_{j = -n..n} (6*k + 5 + 6*j)) ) for n = 0,1,2,.... Cf. A003881. See the example section for notes on the case n = 1.(End)
Equals Product_{p>=5, p prime} p/sqrt(p^2-1). - Dimitris Valianatos, May 13 2017
Equals A019699/4 or A019693/2. - Omar E. Pol, Aug 17 2019
Equals Integral_{x >= 0} (sin(x)/x)^4 = 1/2 + Sum_{n >= 0} (sin(n)/n)^4, by the Abel-Plana formula. - Peter Bala, Nov 05 2019
Equals Integral_{x=0..oo} 1/(1 + x^6) dx. - Bernard Schott, Mar 12 2022
Pi/3 = -Sum_{n >= 1} i/(n*P(n, 1/sqrt(-3))*P(n-1, 1/sqrt(-3))), where i = sqrt(-1) and P(n, x) denotes the n-th Legendre polynomial. The first twenty terms of the series gives the approximation Pi/3 = 1.04719755(06...) correct to 8 decimal places. - Peter Bala, Mar 16 2024
Equals Integral_{x >= 0} (2*x^2 + 1)/((x^2 + 1)*(4*x^2 + 1)) dx. - Peter Bala, Feb 12 2025

A228719 Decimal expansion of 3*Pi/5.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Oct 03 2013

With offset 2, decimal expansion of 6*Pi.
6*Pi is also the surface area of a sphere whose diameter equals the square root of 6. More generally x*Pi is also the surface area of a sphere whose diameter equals the square root of x. - Omar E. Pol, Dec 22 2013
The interior angle of a regular pentagon expressed in radians. - Stefano Spezia, May 30 2025

			3*Pi/5 = 1.8849555921538759430775860299677017305183...
6*Pi = 18.849555921538759430775860299677017305183...

Index entries for transcendental numbers

Cf. A000796, A019692, A122952, A019694, A019669 (Pi through 5*Pi).

Cf. A374172.

First[RealDigits[6*Pi, 10, 100]] (* Paolo Xausa, Jul 16 2024 *)

 6*Pi \\ Charles R Greathouse IV, Sep 30 2022

Equals 1/(10*A374172). - Hugo Pfoertner, Jul 16 2024

A177870 Decimal expansion of 3*Pi/4.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Dec 13 2010

As radians, this is equal to 135 degrees (on an analog clock, the span of 22 minutes and 30 seconds). - Alonso del Arte, Feb 03 2013
Ratio of the area of an arbelos to the area of its associated parbelos. - Jonathan Sondow, Nov 28 2013
(3*Pi/4)*a^2 is the area between a cissoid of Diocles and its asymptote when polar equation of cissoid is r = a*sin^2(t)/cos(t) and Cartesian equation is x * (x^2+y^2) = a * y^2 or y = +-x * sqrt(x/(a-x)). See the curve at the Mathcurve link and formula. - Bernard Schott, Jul 14 2020
The smallest nonnegative solution to sin(x) = -cos(x). - Wolfe Padawer, Apr 12 2023

			2.35619449019234492884698253745962716314787704953132936573120...

Jonathan Borwein & Peter Borwein, A Dictionary of Real Numbers. Pacific Grove, California: Wadsworth & Brooks/Cole Advanced Books & Software (1990) p. 168

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Robert Ferréol, Cissoid of Diocles, Mathcurve.
Jonathan Sondow, The parbelos, a parabolic analog of the arbelos, arXiv:1210.2279 [math.HO], 2012-2013: Amer. Math. Monthly 120 (2013) 929-935.
E. Tsukerman, Solution of Sondow's problem: a synthetic proof of the tangency property of the parbelos, arXiv:1210.5580 [math.MG], 2012-2013.
Index to sequences related to curves
Index entries for transcendental numbers

Reciprocal of A232715.

Cf. A000796, A003881, A019675, A122952.

Maple
```
evalf(3*Pi/4) ;
```
Mathematica
```
RealDigits[N[3(Pi/4), 110]][[1]]
```

3*Pi/4 \\ Charles R Greathouse IV, Sep 30 2022

Equals (3/4)*A000796 = 3*A003881 = 6*A019675 = A122952/4.
Equals 1 + (3/5) + (3*4)/(5*7) + (3*4*5)/(5*7*9) + ... = hypergeom([3,1],[5/2],1/2). - Peter Bala, Oct 30 2019
Equals 2 * Integral_{x=0..1} x * sqrt(x/(1-x)) dx (cissoid). - Bernard Schott, Jul 14 2020
Equals Sum_{k>=1} arctan(2/k^2). - Amiram Eldar, Aug 10 2020
Equals Integral_{x=-oo..oo} (x^2 + 1)/(x^4 + 4) dx. - Kritsada Moomuang, Jun 04 2025

A228824 Decimal expansion of 4*Pi/5.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Oct 03 2013

With offset 2, decimal expansion of 8*Pi.

8*Pi is also the surface area of a sphere whose diameter equals the square root of 8, hence its radius equals the square root of 2. More generally x*Pi is also the surface area of a sphere whose diameter equals the square root of x. - Omar E. Pol, Dec 18 2013

			4*Pi/5 = 2.5132741228718345907701147066236023073577...
8*Pi = 25.132741228718345907701147066236023073577...

Index entries for transcendental numbers

Cf. A002193, A228819.

Cf. A000796, A019692, A122952, A019694, A019669, A228719, A228721 (Pi through 7*Pi).

RealDigits[(4 Pi)/5,10,120][[1]] (* Harvey P. Dale, Aug 24 2017 *)

 8*Pi \\ Charles R Greathouse IV, Oct 01 2022

A336266 Decimal expansion of (3/16)*Pi.

Original entry on oeis.org

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

Offset: 0

Bernard Schott, Jul 15 2020

Area of one egg of the "double egg" whose polar equation is r(t) = a * cos(t)^2 and a Cartesian equation is (x^2+y^2)^3 = a^2*x^4 is equal to (3/16)*Pi * a^2. See the curve at the Mathcurve link.

			0.58904862254808623221174563436490679078696926...

L. Lorentzen and H. Waadeland, Continued Fractions with Applications, North-Holland 1992, p. 586.

Peter Bala, A note on A336266
Robert Ferréol, Double egg, Mathcurve.
Index to sequences related to curves.
Index entries for transcendental numbers

Cf. A259830 (length of an egg).

Cf. A019692 (2*Pi for deltoid), A122952 (3*Pi for cycloid and nephroid), A180434 (2-Pi/2 for Newton strophoid), A197723 (3*Pi/2 for cardioid), A336308.

Maple
```
evalf(3*Pi/16,140);
```

RealDigits[3*Pi/16, 10, 100][[1]] (* Amiram Eldar, Jul 15 2020 *)

PARI
```
3*Pi/16 \\ Michel Marcus, Jul 15 2020
```

Equals Integral_{t=0..Pi} (1/2) * cos(t)^4 * dt.
Equals Integral_{x=0..oo} 1/(x^2 + 1)^3 dx. - Amiram Eldar, Aug 13 2020
From Peter Bala, Mar 21 2024: (Start)
Equals 1/2 + Sum_{n >= 0} (-1)^n/(u(n)*u(-n)), where the polynomial u(n) = (2*n - 1)*(4*n^2 - 4*n + 3)/3 = A057813(n-1) has its zeros on the vertical line Re(z) = 1/2 in the complex plane. Cf. A336308.
Equals 1/2 + 1/(11 + 3/(12 + 15/(12 + 35/(12 + ... + (4*n^2 - 1)/(12 + ... ))))). See Lorentzen and Waadeland, p. 586, equation 4.7.10 with n = 2. (End)

A228721 Decimal expansion of 7*Pi.

Original entry on oeis.org

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

Offset: 2

Omar E. Pol, Oct 03 2013

7*Pi is also the surface area of a sphere whose diameter equals the square root of 7. More generally x*Pi is also the surface area of a sphere whose diameter equals the square root of x. - Omar E. Pol, Dec 22 2013

			21.99114857512855266923850368295652018938...

Index entries for transcendental numbers

Cf. A000796, A019692, A122952, A019694, A019669, A228719 (Pi through 6*Pi).

RealDigits[7*Pi,10,120][[1]] (* Harvey P. Dale, Jan 14 2015 *)

7*Pi \\ Charles R Greathouse IV, Oct 01 2022

A336308 Decimal expansion of (5/32)*Pi.

Original entry on oeis.org

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

Offset: 0

Bernard Schott, Jul 17 2020

(5*Pi/32)*a^2 is the area of a simple folium also called ovoid, or Kepler egg whose polar equation is r = a*cos^3(t) and Cartesian equation is (x^2+y^2)^2 = a * x^3. See the curve at the Mathcurve link.

			0.4908738521234051935097880286374223256558077186523602...

Robert Ferréol, Simple folium, Mathcurve.
Index to sequences related to curves.
Index entries for transcendental numbers

Cf. A019692 (2*Pi for deltoid), A122952 (3*Pi for cycloid and nephroid), A180434 (2-Pi/2 for Newton strophoid), A197723 (3*Pi/2 for cardioid), A336266 (3*Pi/16 for double egg).

Maple
```
evalf(5*Pi/32, 140);
```

RealDigits[5*Pi/32, 10, 100][[1]] (* Amiram Eldar, Jul 17 2020 *)

PARI
```
5*Pi/32 \\ Michel Marcus, Jul 17 2020
```

Equals Integral_{t=0..Pi} cos^6(t)/2 dt (area of simple folium).
From Amiram Eldar, Aug 13 2020: (Start)
Equals Integral_{x=0..oo} 1/(x^2 + 1)^4 dx.
Equals Integral_{x=-1..1} x^3 * arcsin(x) dx. (End)
Equals 5/9 - 10*Sum_{n >= 1} (-1)^(n+1)/(u(n)*u(-n)), where the polynomial u(n) = (2*n - 1)^2*(4*n^2 - 4*n + 9)/3 satisfies the difference equation 16*u(n) = (2*n - 1)*(u(n+1) - u(n-1)) and has its zeros on the vertical line Re(z) = 1/2 in the complex plane. Cf. A336266. - Peter Bala, Mar 25 2024

A229939 Decimal expansion of 9*Pi/10.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Oct 06 2013

With offset 2, decimal expansion of 9*Pi.

9*Pi is also the surface area of a sphere whose diameter equals the square root of 9. More generally x*Pi is also the surface area of a sphere whose diameter equals the square root of x. - Omar E. Pol, Dec 22 2013

			9*Pi/10 = 2.827433388230813914616379044951552595777...
9*Pi = 28.27433388230813914616379044951552595777...

Index entries for transcendental numbers

Cf. A000796, A019692, A122952, A019694, A019669, A228719, A228721, A228824 (Pi through 8*Pi).

First[RealDigits[9*Pi, 10, 100]] (* Paolo Xausa, Jun 18 2024 *)

 9*Pi \\ Charles R Greathouse IV, Oct 01 2022

9*Pi = Sum_{j >= 0} j*(j - 1)*(j - 2)*(j - 3)*2^(j+1) / ((2*j + 1)*binomial(2*j, j)). - Peter Bala, Nov 21 2023

cp's OEIS Frontend

A019669 Decimal expansion of Pi/2.

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A019670 Decimal expansion of Pi/3.

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A228719 Decimal expansion of 3*Pi/5.

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A177870 Decimal expansion of 3*Pi/4.

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A228824 Decimal expansion of 4*Pi/5.

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A336266 Decimal expansion of (3/16)*Pi.

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