A019669 -id:A019669 - cp's OEIS frontend

A137914 Decimal expansion of arccos(1/3).

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

Offset: 1

Rick L. Shepherd, Feb 22 2008

Dihedral angle in radians of regular tetrahedron.
Arccos(1/3) is the central angle of a cube, made by the center and two neighboring vertices. - Clark Kimberling, Feb 10 2009
Also the complementary tetrahedral angle, Pi-A156546, and therefore related to the magic angle (Pi-2*A195696). - Stanislav Sykora, Jan 23 2014
Polar angle (or apex angle) of the cone that subtends exactly one third of the full solid angle. - Stanislav Sykora, Feb 20 2014
Also the acute angle in the rhombi and isosceles trapezoids in the trapezo-rhombic dodecahedron. - Eric W. Weisstein, Jan 09 2019
Also the angle between the tangent lines to the curves y = sin(x) at y = cos(x) at the points of intersection. - David Radcliffe, Jan 17 2023

			1.2309594173407746821349291782479873757103400093550948390555483336639923144...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 58.
Eric Weisstein's World of Mathematics, Tetrahedron.
Eric Weisstein's World of Mathematics, Dihedral Angle.
Eric Weisstein's World of Mathematics, Trapezo-Rhombic Dodecahedron.
Index entries for transcendental numbers

Cf. A137915 (same in degrees), A019670, A195695, A195696, A238238, Platonic solids dihedral angles: A156546 (octahedron), A019669 (cube), A236367 (icosahedron), A137218 (dodecahedron).

SetDefaultRealField(RealField(100)); Arccos(1/3); // G. C. Greubel, Aug 20 2018

RealDigits[ArcCos[1/3], 10, 120][[1]] (* Harvey P. Dale, Jul 06 2018 *)
RealDigits[ArcSec[3], 10, 120][[1]] (* Eric W. Weisstein, Jan 09 2019 *)

PARI
```
acos(1/3)
```

arccos(1/3) = arctan(2*sqrt(2)) = 2*arcsin(sqrt(3)/3) = arcsin(2*sqrt(2)/3).
Equals sqrt(2)*Sum_{k>=0} (-1)^k/(2^k*(2*k+1)). - Davide Rotondo, Jun 07 2025
Equals 2*A195695. - Hugo Pfoertner, Jun 07 2025

A073000 Decimal expansion of arctangent of 1/2.

Original entry on oeis.org

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

Offset: 0

Robert G. Wilson v, Aug 03 2002

The angle at which you must shoot a cue ball on a standard pool table so that it will strike all four sides and return to its origin. [Barrow] - Robert G. Wilson v, Nov 29 2015

			Arctan(1/2)
=0.463647609000806116214256231461214402028537054286120263810933088720197864165... radians
=26°.56505117707798935157219372045329467120421429964522102798601631528806582148474...
=26°33'.9030706246793610943316232271976802722528579787132616791609789172839492890...
=26°33'54".184237480761665659897393631860816335171478722795700749658735037036957...
complement = 63°.43494882292201064842780627954670532879578570035477897201398368471...
supplement = 153°.4349488229220106484278062795467053287957857003547789720139836847...

John D. Barrow, One Hundred Essential Things You Didn't Know You Didn't Know, W. W. Norton & Co., NY & London, 2008.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 242.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Peter Bala, New series for old functions.
R. J. Mathar, Hierarchical Subdivision of the Simple Cubic Lattice, arXiv preprint arXiv:1309.3705 [math.MG], 2013.
Simon Plouffe, arctan(1/2).
Index entries for transcendental numbers

Cf. A001622, A002162, A002391, A105531, A254619.

evalf(arctan(0.5)) ; # R. J. Mathar, Aug 22 2013

Mathematica
```
RealDigits[ ArcTan[1/2], 10, 110] [[1]]
```

default(realprecision,2000); atan(1/2) \\ Anders Hellström, Nov 30 2015

Equals Pi/2 - A105199 = A019669-A105199. - R. J. Mathar, Aug 21 2013
From Peter Bala, Feb 04 2015: (Start)
Arctan(1/2) = 1/2*Sum_{k >= 0} (-1)^k/((2*k + 1)*4^k).
Define a pair of integer sequences A(n) = 4^n*(2*n + 1)!/n! and B(n) = A(n)*Sum_{k = 0..n} (-1)^k/((2*k + 1)*4^k). Both sequences satisfy the same second order recurrence equation u(n) = (12*n + 10)*u(n-1) + 16*(2*n - 1)^2*u(n-2). From this observation we obtain the continued fraction expansion 2*arctan(1/2) = 1 - 2/(24 + 16*3^2/(34 + 16*5^2/(46 + ... + 16*(2*n - 1)^2/((12*n + 10) + ...)))). See A002391, A105531 and A002162 for similar expansions.
Arctan(1/2) = 2/5 * Sum_{k >= 0} (4/5)^k/((2*k + 1)*binomial(2*k,k)).
Define a pair of integer sequences C(n) = 5^n*(2*n + 1)!/n! and D(n) = C(n)*Sum_{k = 0..n} (4/5)^k/((2*k + 1)*binomial(2*k,k)). Both sequences satisfy the same second order recurrence equation u(n) = (24*n + 10)*u(n-1) - 40*n*(2*n - 1)^2*u(n-2). From this observation we obtain the continued fraction expansion 5/2*arctan(1/2) = 1 + 4/(30 - 240/(58 - 600/(82 - ... - 40*n*(2*n - 1)/((24*n + 10) - ... )))).
Arctan(1/2) = 2/25 * Sum_{k >= 0} (24*k + 17)*(4/5)^(2*k)/( (4*k + 1)*(4*k + 3)*binomial(4*k,2*k) ).
Arctan(1/2) = 2/125 * Sum_{k >= 0} (1116*k^2 + 1446*k + 433)*(4/5)^(3*k)/( (6*k + 1)*(6*k + 3)*(6*k + 5)*binomial(6*k,3*k) ). (End)
Equals Integral_{x = 0..oo} exp(-2*x)*sin(x)/x dx. - Peter Bala, Nov 05 2019
Equals 2 * arccot(phi^3), where phi is the golden ratio (A001622). - Amiram Eldar, Jul 06 2023
Equals Sum_{n >= 1} i/(n*P(n, 2*i)*P(n-1, 2*i)) = (1/2)*Sum_{n >= 1} (-1)^(n+1)*4^n/(n*A098443(n)*A098443(n-1)), where i = sqrt(-1) and P(n, x) denotes the n-th Legendre polynomial. The n-th summand of the series is O( 1/(3 + 2*sqrt(2))^n ). - Peter Bala, Mar 16 2024

A014493 Odd triangular numbers.

Original entry on oeis.org

1, 3, 15, 21, 45, 55, 91, 105, 153, 171, 231, 253, 325, 351, 435, 465, 561, 595, 703, 741, 861, 903, 1035, 1081, 1225, 1275, 1431, 1485, 1653, 1711, 1891, 1953, 2145, 2211, 2415, 2485, 2701, 2775, 3003, 3081, 3321, 3403, 3655, 3741, 4005, 4095, 4371, 4465, 4753, 4851

Offset: 1

Mohammad K. Azarian

Odd numbers of the form n*(n+1)/2.
For n such that n(n+1)/2 is odd see A042963 (congruent to 1 or 2 mod 4).
Even central polygonal numbers minus 1. - Omar E. Pol, Aug 17 2011
Odd generalized hexagonal numbers. - Omar E. Pol, Sep 24 2015

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 68.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
D. H. Lehmer, Recurrence formulas for certain divisor functions, Bull. Amer. Math. Soc., Vol. 49, No. 2 (1943), pp. 150-156.
Eric Weisstein's World of Mathematics, Triangular Number.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000217, A000796, A014494, A019669, A042963, A067589, A128880.

List([1..50], n -> (2*n-1)*(2*n-1-(-1)^n)/2); # G. C. Greubel, Feb 09 2019

[(2*n-1)*(2*n-1-(-1)^n)/2: n in [1..50]]; // Vincenzo Librandi, Aug 18 2011

[(2*n-1)*(2*n-1-(-1)^n)/2$n=1..50]; # Muniru A Asiru, Mar 10 2019

Select[ Table[n(n + 1)/2, {n, 93}], OddQ[ # ] &] (* Robert G. Wilson v, Nov 05 2004 *)
LinearRecurrence[{1,2,-2,-1,1},{1,3,15,21,45},50] (* Harvey P. Dale, Jun 19 2011 *)

a(n)=(2*n-1)*(2*n-1-(-1)^n)/2 \\ Charles R Greathouse IV, Sep 24 2015

def A014493(n): return ((n<<1)-1)*(n-(n&1^1)) # Chai Wah Wu, Feb 12 2023

[(2*n-1)*(2*n-1-(-1)^n)/2 for n in (1..50)] # G. C. Greubel, Feb 09 2019

From Ant King, Nov 17 2010: (Start)
a(n) = (2*n-1)*(2*n - 1 - (-1)^n)/2.
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5). (End)
G.f.: x*(1 + 2*x + 10*x^2 + 2*x^3 + x^4)/((1+x)^2*(1-x)^3). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 10 2009
a(n) = A000217(A042963(n)). - Reinhard Zumkeller, Feb 14 2012, Oct 04 2004
a(n) = A193868(n) - 1. - Omar E. Pol, Aug 17 2011
Let S = Sum_{n>=0} x^n/a(n), then S = Q(0) where Q(k) = 1 + x*(4*k+1)/(4*k + 3 - x*(2*k+1)*(4*k+3)^2/(x*(2*k+1)*(4*k+3) + (4*k+5)*(2*k+3)/Q(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Feb 27 2013
E.g.f.: (2*x^2+x+1)*cosh(x)+x*(2*x-1)*sinh(x)-1. - Ilya Gutkovskiy, Apr 24 2016
Sum_{n>=1} 1/a(n) = Pi/2 (A019669). - Robert Bilinski, Jan 20 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2). - Amiram Eldar, Mar 06 2022

More terms from Erich Friedman

A156648 Decimal expansion of Product_{k>=1} (1 + 1/k^2).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Feb 12 2009

Consider the value at s = 2 of the partition zeta functions zeta_{type}(s), where the defining sum runs over partitions into 'type' parts, where 'type' is 'even', 'prime' or 'distinct'. (For the precise definitions see R. Schneider's dissertation.) Then
  zeta_{even}(2) = Pi/2 = A019669;
  zeta_{prime}(2) = Pi^2/6 = A013661;
  zeta_{distinct}(2) = sinh(Pi)/Pi, this constant. - Peter Luschny, Aug 11 2021
For m>0, Product_{k>=1} (1 + m/k^2) = sinh(Pi*sqrt(m)) / (Pi*sqrt(m)). - Vaclav Kotesovec, Aug 30 2024

			3.676077910374977720695697492028260666507156346827630277478003593557447324111... = (1+1)*(1+1/4)*(1+1/9)*(1+1/16)*(1+1/25)*...

Reinhold Remmert, Classical topics in complex function theory, Vol. 172 of Graduate Texts in Mathematics, p. 12, Springer, 1997.

Robert Schneider, Eulerian series, zeta functions and the arithmetic of partitions, arXiv:2008.04243 [math.NT], 2020.

Square root of A084243.

Cf. A084243, A073017, A258870, A258871, A334411, A019669, A013661.

Maple
```
evalf(sinh(Pi)/Pi) ;
```

RealDigits[Sinh[Pi]/Pi, 10, 111][[1]] (* or *)
RealDigits[Re[1/(I!*(-I)!)], 10, 111][[1]] (* Robert G. Wilson v, Feb 23 2015 *)

sinh(Pi)/Pi \\ Charles R Greathouse IV, Dec 16 2013

prodnumrat(1 + 1/x^2, 1) \\ Charles R Greathouse IV, Feb 03 2025

Equals sinh(Pi)/Pi.
Equals 1/A090986. - R. J. Mathar, Mar 05 2009
Binomial(2, 1+i) = 1/(i!*(-i)!) (where x! means Gamma(x+1)). - Robert G. Wilson v, Feb 23 2015
Equals exp(Sum_{j>=1} (-(-1)^j*Zeta(2*j)/j)). - Vaclav Kotesovec, Mar 28 2019
Equals Product_{k>=1} (1+2/(k*(k+2))). - Amiram Eldar, Aug 16 2020

A156546 Decimal expansion of the central angle of a regular tetrahedron.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Feb 09 2009

If O is the center of a regular tetrahedron ABCD, then the central angle AOB is this number; exact value is Pi - arccos(1/3).
The (minimal) central angle of the other four regular polyhedra are as follows:
- cube: A137914,
- octahedron: A019669,
- dodecahedron: A156547,
- icosahedron: A105199.
Dihedral angle of two adjacent faces of the octahedron. - R. J. Mathar, Mar 24 2012
Best known as "tetrahedral angle" theta (e.g., in chemistry). Its Pi complement (i.e., Pi - theta) is the dihedral angle between adjacent faces in regular tetrahedron. - Stanislav Sykora, May 31 2012
Also twice the magic angle (A195696). - Stanislav Sykora, Nov 14 2013

			Pi - arccos(1/3) = 1.910633236249018556..., or, in degrees, 109.471220634490691369245999339962435963006843100... = A247412

Wikipedia, Tetrahedron
Wikipedia, Tetrahedral molecular geometry
Index entries for transcendental numbers.

Cf. A195696, A247412.

Cf. Platonic solids dihedral angles: A137914 (tetrahedron), A019669 (cube), A236367 (icosahedron), A137218 (dodecahedron). - Stanislav Sykora, Jan 23 2014

evalf(arccos(-1/3)) ; # R. J. Mathar, Feb 18 2025

RealDigits[Pi-ArcCos[1/3],10,120][[1]] (* Harvey P. Dale, Aug 25 2011 *)

acos(-1/3) \\ Charles R Greathouse IV, Aug 30 2013

Start with vertices (1,1,1), (1,-1,-1,), (-1,1,-1), and (1,-1,1) and apply the formula for cosine of the angle between two vectors.
Equals 2* A195696. - R. J. Mathar, Mar 24 2012
Equals A000796 - A137914 = A247412 / A072097 - R. J. Mathar, Feb 18 2025

A197723 Decimal expansion of (3/2)*Pi.

Original entry on oeis.org

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

Offset: 1

Alonso del Arte, Oct 17 2011

As radians, this is equal to 270 degrees or 300 gradians.
Multiplying a number by -i (with i being the imaginary unit sqrt(-1)) is equivalent to rotating it by this number of radians on the complex plane.
Named 'Pau' by Randall Munroe, as a humorous compromise between Pi and Tau. - Orson R. L. Peters, Jan 08 2017
(3*Pi/2)*a^2 is the area of the cardioid whose polar equation is r = a*(1+cos(t)) and whose Cartesian equation is (x^2+y^2-a*x)^2 = a^2*(x^2+y^2). The length of this cardioid is 8*a. See the curve at the Mathcurve link. - Bernard Schott, Jan 29 2020

			4.712388980384689857693965074919254326296...

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Robert Ferréol, Cardioid, Mathcurve.
Randall Munroe, xkcd: Pi vs. Tau
Eric Weisstein's World of Mathematics, Cardioid
Index entries for transcendental numbers

Cf. A019669, A003881, A347152.

Digits:=100: evalf(3*Pi/2); # Wesley Ivan Hurt, Jan 08 2017

Mathematica
```
RealDigits[3Pi/2, 10, 105][[1]]
```

3*Pi/2 \\ Charles R Greathouse IV, Jul 06 2018

2*Pi - Pi/2 = Pi + Pi/2.
Equals Integral_{t=0..Pi} (1+cos(t))^2 dt. - Bernard Schott, Jan 29 2020
Equals -4 + Sum_{k>=1} (k+1)*2^k/binomial(2*k,k). - Amiram Eldar, Aug 19 2020

A019679 Decimal expansion of Pi/12.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Equals cone's volume (radius = 1/2, height = 1) and semi-sphere's volume (radius = 1/2). - Eric Desbiaux, Dec 08 2008
Decimal expansion of least x > 0 having cos(4x) = (cos 3x)^2. See A197476. - Clark Kimberling, Oct 15 2011
Multiplied by 10, decimal expansion of 5*Pi/6. - Alonso del Arte, Aug 19 2013
Volume between a cylinder and the inscribed sphere of diameter 1. - Omar E. Pol, Sep 25 2013

			Pi/12 = 0.2617993877991494365385536152732919070164307...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.4, p. 492.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Index entries for transcendental numbers.

Cf. A000796, A003881, A019669, A019670, A019675, A019683, A244978.

RealDigits[N[Pi/12, 6! ]] (* Vladimir Joseph Stephan Orlovsky, Dec 02 2009 *)
RealDigits[Pi/12,10,120][[1]] (* Harvey P. Dale, Jan 12 2024 *)

Pi/12 \\ Charles R Greathouse IV, Sep 28 2022

A003881 - A019673. - Omar E. Pol, Sep 25 2013
Equals Integral_{x = 0..1} x^2*sqrt(1 - x^6) dx. - Peter Bala, Oct 27 2019
Equals Sum_{k>=0} binomial(2*k,k)/((2*k+1)*4^(2*k+1)). - Amiram Eldar, May 30 2021
Constant divided by 10 = Pi/120 = 0.0261799387... =  Sum_{n = -oo..oo} 1/((4*n+1)*(4*n+2)*(4*n+3)*(4*n+5)*(4*n+6)*(4*n+7)) (using the Eisenstein summation convention Sum_{n = -oo..oo} = lim_{N -> oo} Sum_{n = -N..N}). Note that 22/7 - Pi = 240*Sum_{n >= 1} 1/((4*n+1)*(4*n+2)*(4*n+3)*(4*n+5)*(4*n+6)*(4*n+7)). - Peter Bala, Nov 28 2021

A096444 Decimal expansion of (Pi - 1)/2.

Original entry on oeis.org

1, 0, 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, 3, 9, 9, 1, 0, 7

Offset: 1

N. J. A. Sloane, Aug 16 2004

From Bernard Schott, Apr 19 2021: (Start)
The series Sum_{k>=1} sin(k)/k and also Sum_{k>=1} cos(k)/k (A121225) are called Fresnel series.
The series Sum_{k>=1} |sin(k)/k| is divergent. (End)

			1.0707963267948966...

Xavier Merlin, Methodix Analyse, Ellipses, 1997, p. 117.

Ira Rosenholtz, Tangent sequences, world records, Pi, and the meaning of life: Some Applications of Number Theory to Calculus, Math. Mag., Vol. 72, No. 5 (1999), pp. 367-376.
Jan W. H. Swanepoel, On a generalization of a theorem by Euler, Journal of Number Theory, Vol. 149 (April 2015), pp. 46-56.
Université de Rennes, Séries semi-convergentes, Base raisonnée d'exercices de Mathématiques, p. 2 (Braise).
Index entries for transcendental numbers.

Cf. A019669, A096418, A121225, A342680.

pi:=Pi(RealField(110)); Reverse(Intseq(Floor(10^105*(pi-1)/2))); // Vincenzo Librandi, Mar 12 2015

RealDigits[(Pi - 1)/2, 10, 105][[1]] (* Robert G. Wilson v, Aug 26 2004 *)

(Pi-1)/2 \\ Charles R Greathouse IV, Jan 30 2015

Equals Sum_{k >= 1} sin(k)/k. (This follows from the identity x = Pi - 2 Sum_{k >= 1} sin(k*x)/k, as observed by Euler in 1744.)
Equals A019669 minus 1/2. - R. J. Mathar, Dec 15 2008
Equals Sum_{k >= 1} (sin(k)/k)^2. (Interestingly, Sum_{k >= 1} sin(k)/k = Sum_{k >= 1} (sin(k)/k)^2, a series whose terms sum to the sum of the square of each term.) - Dimitri Papadopoulos, Mar 11 2015
Equals arctan(sin(1)/(1-cos(1))). - Amiram Eldar, Jun 06 2021

More terms from Robert G. Wilson v, Aug 17 2004

Better definition from Eric W. Weisstein, Aug 18 2004

A256853 Decimal expansion of the area of a unit 9-gon.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Apr 12 2015

From Michal Paulovic, May 09 2024: (Start)
This constant multiplied by the square of the side length of a regular enneagon equals the area of that enneagon.
9^2 divided by this constant equals 36 * tan(Pi/9) = 13.10292843... which is the perimeter and the area of an equable enneagon with its side length 4 * tan(Pi/9) = 1.45588093... . (End)

			6.181824193772900127213744059619763614941713348134358098386864...

Stanislav Sykora, Table of n, a(n) for n = 1..2000
Wikipedia, Nonagon
Wikipedia, Regular polygon
Index entries for algebraic numbers, degree 6

Cf. A000796, A019669, A019670, A019673, A019676, A019685, A019968, A120011 (p=3), A102771 (p=5), A104956 (p=6), A178817 (p=7), A090488 (p=8), A178816 (p=10), A256854 (p=11), A178809 (p=12).

evalf(9 / (4 * tan(Pi/9)), 100); # Michal Paulovic, May 09 2024

RealDigits[(9/4)*Cot[Pi/9], 10, 50][[1]] (* G. C. Greubel, Jul 03 2017 *)

p=9; a=(p/4)*cotan(Pi/p)        \\ Use realprecision in excess

Equals (p/4)*cot(Pi/p), with p = 9.
From Michal Paulovic, May 09 2024: (Start)
Equals 9 * sqrt(2 / (1 - sin(5 * A000796 / 18)) - 1) / 4.
Equals 9 * sqrt(2 / (1 - sin(5 * A019669 / 9)) - 1) / 4.
Equals 9 * sqrt(2 / (1 - sin(5 * A019670 / 6)) - 1) / 4.
Equals 9 * sqrt(2 / (1 - sin(5 * A019673 / 3)) - 1) / 4.
Equals 9 * sqrt(2 / (1 - sin(5 * A019676 / 2)) - 1) / 4.
Equals 9 * sqrt(2 / (1 - sin(50 * A019685)) - 1) / 4.
Equals 9 * sqrt(2 / (1 - sin(5 * Pi / 18)) - 1) / 4.
Equals 9 * sqrt(4 / (2 - i^(4/9) - i^(-4/9)) - 1) / 4.
Equals 9 * sqrt(1 / (8 - (-32 + sqrt(-3072))^(1/3) - (-32 - sqrt(-3072))^(1/3)) - 1/16). (End)
Largest of the 6 real-valued roots of 4096*x^6 -186624*x^4 +1154736*x^2 -177147 =0. - R. J. Mathar, Aug 29 2025

A122952 Decimal expansion of 3*Pi.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Sep 30 2006

Area of the unit cycloid with cusp at the origin, whose parametric formula is x = t - sin(t) and y = 1 - cos(t).
The arc length Integral_{theta=0..2*Pi} sqrt(2(1-cos(theta))) (d theta) = 8.
3*Pi is also the surface area of a sphere whose diameter equals the square root of 3. 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
3*Pi is also the area of the nephroid (an epicycloid with two cusps) whose Cartesian parametrization is: x = (1/2) * (3*cos(t) - cos(3t)) and y = (1/2) * (3*sin(t) - sin(3t)). The length of this nephroid is 12. See the curve at the Mathcurve link. - Bernard Schott, Feb 01 2020

			9.424777960769379715387930149838508652591508198125317462924833776...

Anton, Bivens & Davis, Calculus, Early Transcendentals, 7th Edition, John Wiley & Sons, Inc., NY 2002, p. 490.
William H. Beyer, Editor, CRC St'd Math. Tables, 27th Edition, CRC Press, Inc., Boca Raton, FL, 1984, p. 214.

Robert Ferréol, Nephroid, Mathcurve.
Eric Weisstein's World of Mathematics, Cycloid.
Index entries for transcendental numbers

Cf. A000796, A019692, A019694, A019669.

Cf. A093828, A180434, A197723.

Mathematica
```
RealDigits[3Pi, 10, 111][[1]]
```

3*Pi \\ Charles R Greathouse IV, Sep 28 2022

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