A188615 -id:A188615 - cp's OEIS frontend

A244644 Consider the method used by Archimedes to determine the value of Pi (A000796). This sequence denotes the number of iterations of his algorithm which would result in a difference of less than 1/10^n from that of Pi.

0, 1, 3, 5, 6, 8, 10, 11, 13, 15, 16, 18, 20, 21, 23, 25, 26, 28, 29, 31, 33, 34, 36, 38, 39, 41, 43, 44, 46, 48, 49, 51, 53, 54, 56, 58, 59, 61, 63, 64, 66, 68, 69, 71, 73, 74, 76, 78, 79, 81, 83, 84, 86, 88, 89, 91, 93, 94, 96, 98, 99, 101, 103, 104, 106, 108, 109, 111, 113, 114

Offset: 0

William H. Richardson and Robert G. Wilson v, Jul 03 2014

It takes on average 5/3 iterations to yield another digit in the decimal expansion of Pi.
The side of a 96-gon inscribed in a unit circle is equal to sqrt(2-sqrt(2+sqrt(2+sqrt(2+sqrt(3))))). This is the size of one of the two polygons that Archimedes used to derive that 3 + 10/70 < Pi < 3 + 10/71.
In the Mathematica program, we started with an inscribed triangle and a circumscribed triangle of a unit circle and used decimal precision to just over a 1000 places.
The perimeter of the circumscribed 3*2^n-polygon exceeds Pi by more than the deficit of the perimeter of the inscribed 3*2^n-polygon. If we were to give twice the weight of the inscribed 3*2^n-polygon to that of the circumscribed 3*2^n-polygon, then the convergence would be twice as fast!
From A.H.M. Smeets, Jul 12 2018: (Start)
Archimedes's scheme: set upp(0) = 2*sqrt(3), low(0) = 3 (hexagons); upp(n+1) = 2*upp(n)*low(n)/(upp(n)+low(n)) (harmonic mean, i.e., 1/upp(n+1) = (1/upp(n) + 1/low(n))/2), low(n+1) = sqrt(upp(n+1)*low(n)) (geometric mean, i.e., log(low(n+1)) = (log(upp(n+1)) + log(low(n)))/2), for n >= 0. Invariant: low(n) < Pi < upp(n); variant function: upp(n)-low(n) tends to zero for n -> inf. The error of low(n) and upp(n) decreases by a factor of approximately 4 each iteration, i.e., approximately 2 bits are gained by each iteration.
From Archimedes's scheme, set r(n) = (2*low(n) + upp(n))/3, r(n) > Pi and the error decreases by a factor of approximately 16 for each iteration, i.e., approximately 4 bits are gained by each iteration. This is often called "Snellius acceleration".
For similar schemes see also A014549 (in this case with quadratically convergence), A093954, A129187, A129200, A188615, A195621, A202541.
Note that replacing "5/3" by "log(10)/log(4)" would be better in the first comment. (End)

			Just averaging the initial two triangles (3.89711) would yield Pi to one place of accuracy, i.e., the single digit '3'. Therefore a(0) = 0.
The first iteration yields, as the perimeters of the two hexagons, 4*sqrt(3) and 6. Their average is ~ 3.2320508 which is within 1/10 of the true value of Pi. Therefore a(1) = 1.
a(3) = 5 since it takes 5 iterations of Archimedes's algorithm to drive the averaged value of the circumscribed 96-gon and the inscribed 96-gon to yield a value within 0.001 of the correct value of Pi.
a(4) = 6 since it takes 6 iterations of Archimedes's algorithm to drive the averaged value of the circumscribed 3*2^6-gon and the inscribed 3*2^6-gon to yield a value within 0.0001 of the correct value of Pi.

Petr Beckmann, A History of Pi, 5th Ed. Boulder, Colorado: The Golem Press (1982).
Jonathan Borwein and David Bailey, Mathematics by Experiment, Second Edition, A. K. Peters Ltd., Wellesley, Massachusetts 2008.
Jonathan Borwein & Keith Devlin, The Computer As Crucible, An Introduction To Experimental Mathematics, A. K. Peters, Ltd., Wellesley, MA, Chapter 7, 'Calculating [Pi]' pp. 71-79, 2009.
Eli Maor, The Pythagorean Theorem, Princeton Science Library, Table 4.1, page 55.
Daniel Zwillinger, Editor-in-Chief, CRC Standard Mathematical Tables and Formulae, 31st Edition, Chapman & Hall/CRC, Boca Raton, London, New York & Washington, D.C., 2003, §4.5 Polygons, page 324.

Mike Bertrand, Ex Libris, Archimedes and Pi
Frits Beukers and Weia Reinboud, Snellius versneld, (text in English), preprint.
Frits Beukers and Weia Reinboud, Snellius versneld, (text in English), NAW 5/3 no. 1, pp. 60-63 (2002).
Lee Fook Loong Eugene, The Computation of [Pi] And Its History
Kyutae Paul Han, Pi and Archimedes Polygon Method
Eric Weisstein's World of Mathematics, Archimedes' Recurrence Formula
Eric Weisstein's World of Mathematics, Regular Polygon
Michael Woltermann Ph.D., Washington & Jefferson College, 38. Archimedes' Determination of [Pi].

Cf. A000796.

a[n_] := a[n] = N[2 a[n - 1] b[n - 1]/(a[n - 1] + b[n - 1]), 2^10]; b[n_] := b[n] = N[ Sqrt[ b[n - 1] a[n]], 2^10]; a[-1] = 2Sqrt[27]; b[-1] = a[-1]/2; f[n_] := Block[{k = 0}, While[ 10^n*((a[k] + b[k])/4 -Pi) > 1, k++]; k]; Array[f, 70]

Conjecture: There exists a c such that a(n) = floor(n*log(10)/log(4)+c); where c is in the range [0.08554,0.10264]. Critical values to narrow the range are believed to be at a(74), a(133), a(192), a(251), a(310), a(366), a(425), a(484). - A.H.M. Smeets, Jul 23 2018

A188595 Decimal expansion of Brocard angle of side-golden right triangle.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Apr 05 2011

The Brocard angle is invariant of the size of the side-golden right triangle ABC. The shape of ABC is given by sidelengths a,b,c, where a=r*b, and c=sqrt(a^2+b^2), where r=(golden ratio)=(1+sqrt(5))/2. This is the unique right triangle matching the continued fraction [1,1,1,...] of r; i.e, under the side-partitioning procedure described in the 2007 reference, there is exactly 1 removable subtriangle at each stage. (This is analogous to the removal of 1 square at each stage of the partitioning of the golden rectangle as a nest of squares.)

Also <3_5> in Conway et al. (1999). - Eric W. Weisstein, Nov 06 2024

			Brocard angle: 0.420534335283965127888262515913215373 approx.

G. C. Greubel, Table of n, a(n) for n = 0..5000
John H. Conway, Charles Radin, and Lorenzo Sadun, On Angles Whose Squared Trigonometric Functions are Rational, arXiv:math-ph/9812019, 1998. See also Discr. Computat. Geom. (1999) Vol. 22, 321-332.
Eric Weisstein's World of Mathematics, Dehn Invariant.
Index entries for transcendental numbers.

Cf. A188594, A152149, A188615, A228496.

[Arctan(Sqrt(1/5))]; // G. C. Greubel, Nov 21 2017

r=(1+5^(1/2))/2; b=1; a=r*b; c=(a^2+b^2)^(1/2); area=(1/4)((a+b+c)(b+c-a)(c+a-b)(a+b-c))^(1/2); brocard = ArcCot[(a^2+b^2+c^2)/(4 area)]; RealDigits[N[brocard,130]][[1]]
RealDigits[ArcTan[Sqrt[1/5]], 10, 50][[1]] (* G. C. Greubel, Nov 21 2017 *)

atan(sqrt(1/5)) \\ G. C. Greubel, Nov 21 2017

Brocard angle: arccot((a^2+b^2+c^2)/(4*area(ABC))) = arccot(sqrt(5)).

Equals A228496/2. - Hugo Pfoertner, Nov 06 2024

A195699 Decimal expansion of arcsin(sqrt(1/8)) and of arccos(sqrt(7/8)).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Sep 23 2011

			arcsin(sqrt(1/8)) = 0.3613671239067078055891886763206666...

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for transcendental numbers

Cf. A195700, A195703, A195704.

[Arcsin(Sqrt(1/8))]; // G. C. Greubel, Nov 18 2017

r = Sqrt[1/8];
N[ArcSin[r], 100]
RealDigits[%]  (* A195699 *)
N[ArcCos[r], 100]
RealDigits[%]  (* A168229 *)
N[ArcTan[r], 100]
RealDigits[%]  (* A188615 *)
N[ArcCos[-r], 100]
RealDigits[%]  (* A195704 *)

asin(sqrt(1/8)) \\ G. C. Greubel, Nov 18 2017

From Peter Bala, Jan 14 2022: (Start)
Equals (1/2)*arccos(3/4) = arctan(sqrt(7)/7).
Equals sqrt(7)*Sum_{n >= 0} 1/((16*n + 8)*(2^n)*binomial(2*n,n)).
Equals sqrt(2)*Sum_{n >= 0} binomial(2*n,n)/((8*n + 4)*32^n). (End)

A236556 Decimal expansion of the steradian solid angle subtended by one square facet of the cuboctahedron.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Jan 28 2014

Also the vertex solid angle of a regular octahedron.

			1.35934763781648774838557005356705626555297876132983228572769584995966...

Stanislav Sykora, Table of n, a(n) for n = 1..2000
Eric Weisstein's World of Mathematics, Cuboctahedron.
Eric Weisstein's World of Mathematics, Spherical Polygon.
Wikipedia, Platonic solid.
Index entries for transcendental numbers.

Cf. A188615, A236555. Icosidodecahedron: A319881, A319883. Vertex Angles: A019669, A236557, A236558.

evalf(4*arcsin(1/3),100) ; # R. J. Mathar, Apr 26 2021

RealDigits[4*ArcSin[1/3], 10, 120][[1]] (* Amiram Eldar, May 22 2023 *)

PARI
```
4*asin(1/3)
```

Equals 4*arcsin(1/3) = 4*A188615.
Equals 2*Pi - 4*arcsin(2*sqrt(2)/3). - Bradley Klee, Oct 04 2018
8*A236555 + 6*this = 4*Pi. - Bradley Klee, Oct 04 2018

Definition corrected by Bradley Klee, Oct 04 2018

A349580 Decimal expansion of the 5-dimensional Steinmetz solid formed by the intersection of 5 unit-diameter 5-dimensional cylinders whose axes are mutually orthogonal and intersect at a single point.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Nov 22 2021

The constant given by Hildebrand et al. (2012) and Kong et al. (2013) is for unit-radius cylinders, and is thus larger by a factor of 2^5. The constant here, for a unit-diameter cylinders, is analogous to the 3-dimensional case given by Moore (1974).

			0.17198723701328857806510936213684483040318641193634...

A. J. Hildebrand, Lingyi Kong, Abby Turner and Ananya Uppal, Applications of n-dimensional Integrals: Random Points, Broken Sticks and Intersecting Cylinders, Illinois Geometry Lab Project Report, University of Illinois at Urbana-Champaign, December 11, 2012.
Lingyi Kong, Luvsandondov Lkhamsuren, Abigail Turner, Aananya Uppal and A. J. Hildebrand, Intersecting Cylinders: From Archimedes and Zu Chongzhi to Steinmetz and Beyond, Illinois Geometry Lab Project Report, University of Illinois at Urbana-Champaign, April 25, 2013.
Moreton Moore, Symmetrical Intersections of Right Circular Cylinders, The Mathematical Gazette, Vol. 58, No. 405 (1974), pp. 181-185.

Cf. A101465, A188615, A349577, A349578, A349579.

RealDigits[8 * (Pi/12 - ArcCot[2*Sqrt[2]]/Sqrt[2]), 10, 100][[1]]

Equals 8 * (Pi/12 - arccot(2*sqrt(2))/sqrt(2)).

A195704 Decimal expansion of arccos(-sqrt(1/8)).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 23 2011

			arccos(-sqrt(1/8)) = 1.93216345070...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for transcendental numbers

Cf. A195699.

[Arccos(-Sqrt(1/8))]; // G. C. Greubel, Nov 18 2017

r = Sqrt[1/8];
N[ArcSin[r], 100]
RealDigits[%]  (* A195699 *)
N[ArcCos[r], 100]
RealDigits[%]  (* A168229 *)
N[ArcTan[r], 100]
RealDigits[%]  (* A188615 *)
N[ArcCos[-r], 100]
RealDigits[%]  (* A195704 *)

acos(-sqrt(1/8)) \\ G. C. Greubel, Nov 18 2017

Equals Pi - arcsin(sqrt(7/2)/2) = Pi - arctan(sqrt(7)). - Amiram Eldar, Jul 09 2023

A195729 Decimal expansion of arctan(3).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 23 2011

			arctan(3) = 1.2490457723982544258299170772...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for transcendental numbers

Cf. A105531, A137914, A188615.

SetDefaultRealField(RealField(100)); Arctan(3); // G. C. Greubel, Aug 20 2018

r = 3;
N[ArcTan[r], 100]
RealDigits[%]  (* A195729 *)
N[ArcCot[r], 100]
RealDigits[%]  (* A105531 *)
N[ArcSec[r], 100]
RealDigits[%]  (* A137914 *)
N[ArcCsc[r], 100]
RealDigits[%]  (* A188615 *)

atan(3) \\ Charles R Greathouse IV, Sep 23 2014

Equals arctan(1) + arctan(1/2). - Charles R Greathouse IV, Sep 23 2014

Equals arcsin(3/sqrt(10)) = arccos(sqrt(1/10)). - Amiram Eldar, Jul 11 2023

A377203 Decimal expansion of Integral_{x=0..oo} exp(-x)*erf(sqrt(x))^3 dx, where erf is the error function.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Oct 19 2024

			0.458941071620926611833296582700838961085659074877...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Erf.
Wikipedia, Error function.

Cf. A000796, A010474, A188615, A377202.

First[RealDigits[Sqrt[18]*ArcCsc[3]/Pi, 10, 100]]

Equals 3*sqrt(2)*arccot(2*sqrt(2))/Pi = A010474*A188615/A000796 (cf. eq. 38 in Weisstein link).

A349605 Decimal expansion of the probability that the intersection of a cube with random plane that passes through its center is a hexagon.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Nov 23 2021

The normal to the random plane is in the direction from the center of the cube to a point uniformly chosen at random on the surface of a sphere whose center coincides with the center of the cube.

			0.35095931218364362102513335533458546789977189663640...

Su Pernu Mero, Problem 2065, Mathematics Magazine, Vol. 92, No. 1, (2019), p. 73; Solution, by Bill Cowieson, ibid., Vol. 93, No. 1 (2020), pp. 76-77.
Yawen Zhang, A purely 3-D geometrical solution to Mathematics Magazine Problem 2065, arXiv:2010.12349 [math.HO], 2020.

Cf. A137914, A188615.

RealDigits[1 - 6 * ArcSin[1/3] / Pi, 10, 100][[1]]

1 - 6*asin(1/3)/Pi \\ Michel Marcus, Nov 23 2021

Equals 1 - 6*arcsin(1/3)/Pi.

Equals 6*arccos(1/3)/Pi - 2.

cp's OEIS Frontend

A244644 Consider the method used by Archimedes to determine the value of Pi (A000796). This sequence denotes the number of iterations of his algorithm which would result in a difference of less than 1/10^n from that of Pi.

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A188595 Decimal expansion of Brocard angle of side-golden right triangle.

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A195699 Decimal expansion of arcsin(sqrt(1/8)) and of arccos(sqrt(7/8)).

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A236556 Decimal expansion of the steradian solid angle subtended by one square facet of the cuboctahedron.

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A349580 Decimal expansion of the 5-dimensional Steinmetz solid formed by the intersection of 5 unit-diameter 5-dimensional cylinders whose axes are mutually orthogonal and intersect at a single point.

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Mathematica

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A195704 Decimal expansion of arccos(-sqrt(1/8)).

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Magma

Mathematica

PARI

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A195729 Decimal expansion of arctan(3).

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