A003881 -id:A003881 - cp's OEIS frontend

A216546 Decimal expansion of Sum_{k=1..5000000} (-1)^(k-1)/(2k-1).

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

Offset: 0

N. J. A. Sloane, Sep 08 2012

An approximation to Pi/4. The constant is rational, by definition (a product of finitely many rational numbers).

			0.7853981133974483096161608458198756960492923498468264552437...

J. M. Borwein, P. B. Borwein and K. Dilcher, Pi, Euler numbers and asymptotic expansions, Amer. Math. Monthly, 96 (1989), 681-687.
J. M. Borwein and R. M. Corless, Review of "An Encyclopedia of Integer Sequences" by N. J. A. Sloane and Simon Plouffe, SIAM Review, 38 (1996), 333-337.
Index entries for linear recurrences with constant coefficients.

Cf. A003881, A216542, A013706, A216543, A216544, A216545, A013705, A216546, A216547, A216548.

Cf. also A195793.

Digits:=300; M:=5000000; add(evalf((-1)^(k-1)/(2*k-1)), k=1..M);

RealDigits[Total[Table[(-1)^(k-1)/(2k-1),{k,5*10^6}]],10,130][[1]] (* Harvey P. Dale, May 06 2015 *)

A258814 Decimal expansion of the Dirichlet beta function of 7.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 11 2015

			0.9995545078905399094963465498990589830021884819499757922526492189419...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Dirichlet Beta Function.
Wikipedia, Dirichlet beta function.

Cf. A003881 (beta(1)=Pi/4), A006752 (beta(2)=Catalan), A153071 (beta(3)), A175572 (beta(4)), A175571 (beta(5)), A175570 (beta(6)), A258815 (beta(8)), A258816 (beta(9)).

SetDefaultRealField(RealField(100)); R:= RealField(); 61*Pi(R)^7/184320; // G. C. Greubel, Aug 24 2018

RealDigits[DirichletBeta[7], 10, 102] // First

61*Pi^7/184320 \\ Charles R Greathouse IV, Dec 06 2016

beta(7) = Sum_{n>=0} (-1)^n/(2n+1)^7 = (zeta(7, 1/4) - zeta(7, 3/4))/16384 = 61*Pi^7/184320.

Equals Product_{p prime >= 3} (1 - (-1)^((p-1)/2)/p^7)^(-1). - Amiram Eldar, Nov 06 2023

A258816 Decimal expansion of the Dirichlet beta function of 9.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 11 2015

			0.999949684187220089821358873293847527372747996917961601223162723...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Dirichlet Beta Function.
Wikipedia, Dirichlet beta function.

Cf. A003881 (beta(1)=Pi/4), A006752 (beta(2)=Catalan), A153071 (beta(3)), A175572 (beta(4)), A175571 (beta(5)), A175570 (beta(6)), A258814 (beta(7)), A258815 (beta(8)).

SetDefaultRealField(RealField(100)); R:=RealField(); 277*Pi(R)^9/8257536; // G. C. Greubel, Aug 24 2018

RealDigits[DirichletBeta[9], 10, 104] // First

default(realprecision, 100); 277*Pi^9/8257536 \\ G. C. Greubel, Aug 24 2018

beta(9) = Sum_{n>=0} (-1)^n/(2n+1)^9 = (zeta(9, 1/4) - zeta(9, 3/4))/262144 = 277*Pi^9/8257536.

Equals Product_{p prime >= 3} (1 - (-1)^((p-1)/2)/p^9)^(-1). - Amiram Eldar, Nov 06 2023

A084254 Decimal expansion of Sum_{k>=1} 1/(k(exp(2Pi*k)-1)).

Original entry on oeis.org

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

Offset: 0

Benoit Cloitre, Jun 21 2003

			0.00187268244976854611563857947996139886916289565261...

Bruce C. Berndt, Ramanujan Notebook part II, Infinite series, Springer Verlag, 1989, pp. 280-281.

Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020-2022, p. 6.
Simon Plouffe, Identities inspired by Ramanujan Notebooks (part 2), April 2006.
Linas Vepštas, On Plouffe's Ramanujan identities, The Ramanujan Journal, Vol. 27 (2012), pp. 387-408; arXiv preprint, arXiv:math/0609775 [math.NT], 2006-2010.

Cf. A000203, A003881, A068465, A094640.

Cf. A255695 (S(1,1)), A255697 (S(1,4)), A255698 (S(3,1)), A255699 (S(3,2)), A255700 (S(3,4)), A255701 (S(5,1)), A255702 (S(5,2)), A255703 (S(5,4)).

digits = 104; S[1, 2] = NSum[1/(n*(Exp[2*Pi*n] - 1)), {n, 1, Infinity}, WorkingPrecision -> digits+10, NSumTerms -> digits]; RealDigits[S[1, 2], 10, digits] // First (* Jean-François Alcover, Mar 02 2015 *)
Join[{0,0},RealDigits[Log[4/Pi]/4 - Pi/12 + Log[Gamma[3/4]], 10, 100][[1]]] (* Amiram Eldar, May 21 2022 *)

PARI
```
1/4*log(4/Pi)-Pi/12+log(gamma(3/4))
```

Equals log(4/Pi)/4 - Pi/12 + log(Gamma(3/4)).
From Jean-François Alcover, Mar 02 2015: (Start)
This is the case k=1, m=2 of the Plouffe sum S(k,m) = Sum_{n >= 1} 1/(n^k*(exp(m*Pi*n)-1)).
Pi = 72*S(1,1) - 96*S(1,2) + 24*S(1,4). (End)
Equals Sum_{k>=1} sigma(k)/(k*exp(2*Pi*k)). - Amiram Eldar, Jun 05 2023

A258815 Decimal expansion of the Dirichlet beta function of 8.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 11 2015

			0.99984999024682965633806705924046378147600743300742806972498742924...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Dirichlet Beta Function.
Wikipedia, Dirichlet beta function.

Cf. A003881 (beta(1)=Pi/4), A006752 (beta(2)=Catalan), A153071 (beta(3)), A175572 (beta(4)), A175571 (beta(5)), A175570 (beta(6)), A258814 (beta(7)), A258816 (beta(9)).

RealDigits[DirichletBeta[8], 10, 102] // First

(zetahurwitz(8,1/4)-zetahurwitz(8,3/4))*(1/4)^8 \\ Hugo Pfoertner, Feb 07 2020

beta(8) = Sum_{n>=0} (-1)^n/(2n+1)^8 = (zeta(8, 1/4) - zeta(8, 3/4))/65536 = (PolyGamma(7, 1/4) - PolyGamma(7, 3/4))/330301440.
Equals ClausenFunction(8, Pi/2).
Equals Product_{p prime >= 3} (1 - (-1)^((p-1)/2)/p^8)^(-1). - Amiram Eldar, Nov 06 2023

A014200 Number of solutions to x^2 + y^2 <= n, excluding (0,0), divided by 4.

Original entry on oeis.org

0, 1, 2, 2, 3, 5, 5, 5, 6, 7, 9, 9, 9, 11, 11, 11, 12, 14, 15, 15, 17, 17, 17, 17, 17, 20, 22, 22, 22, 24, 24, 24, 25, 25, 27, 27, 28, 30, 30, 30, 32, 34, 34, 34, 34, 36, 36, 36, 36, 37, 40, 40, 42, 44, 44, 44, 44, 44, 46

Offset: 0

N. J. A. Sloane

nonn

From Ant King, Mar 15 2013: (Start)
The terms of this sequence give a running total of the excess of the 4k + 1 divisors of the natural numbers (from 1 through to n) over their 4k + 3 divisors.
To see how good the approximation n * Pi/4 is to a(n), note that a(10^6) = 785387 whereas 10^6 * Pi/4 rounds to 785398. (End)

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A014198, A059851, A101455.

Partial sums of A002654.

1/4*Prepend[SquaresR[2,#]&/@Range[58],0]//Accumulate (* Ant King, Mar 15 2013 *)

a(n) = sum(k=1, n, sumdiv(k, d, kronecker(-4, k/d))); \\ Seiichi Manyama, Dec 18 2021

a(n) = A014198(n) / 4.
Limit_{n->infinity} a(n)/n = Pi/4 = A003881.
a(n) = n - floor(n/3) + floor(n/5) - floor(n/7) + floor(n/9) - floor(n/11) + ... - Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 28 2003
G.f.: (1/(1 - x))*Sum_{k>=1} x^k/(1 + x^(2*k)). - Ilya Gutkovskiy, Dec 23 2016

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

A244854 Decimal expansion of Pi^2/32.

Original entry on oeis.org

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

Offset: 0

Charles R Greathouse IV, Jul 07 2014

Probability that a point selected uniformly at random from the unit 4-cube is in the unit 4-sphere.

Let S(n) = 1 - 1/3 + 1/5 - ... + ((-1)^(n-1))/(2n-1). Then Sum{n >=1} ((-1)^(n-1))*S(n) /(2n+1) = Pi^2 /32. The convergence is very slow. - Michel Lagneau, Feb 27 2015

			Choose -1 <= w, x, y, z <= 1 uniformly at random. Then this constant is the probability that w^2 + x^2 + y^2 + z^2 <= 1.

Index entries for transcendental numbers

Cf. A003881 (2-dimensional analog), A019673 (3-dimensional analog).

Digits:=100; evalf(Pi^2/32); # Wesley Ivan Hurt, Feb 27 2015

RealDigits[Pi^2/32,10,120][[1]] (* Harvey P. Dale, Jul 13 2014 *)

PARI
```
Pi^2/32
```

Equals Integral_{0..infinity} x^2*BesselK(0, x)^2 dx. - Jean-François Alcover, Apr 15 2015
Equals Integral_{x=0..1} arctan(x)/(1+x^2) dx. - Amiram Eldar, Aug 09 2020
Equals Integral_{x=0..1} Integral_{y=0..1} Integral_{z=0..1} (1 + x^2 + y^2 + z^2)^(-2). - Peter Luschny, Dec 10 2022

A308715 Decimal expansion of cosh(Pi/2).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Jun 19 2019

			2.50917847865805678200999564326940594821202435...

Cf. A003881, A016754, A156648, A308716.

Mathematica
```
RealDigits[Cosh[Pi/2], 10, 120][[1]]
```

cosh(Pi/2) \\ Michel Marcus, Jun 20 2019

Equals Product_{k>=1} (1 + 1/(2*k - 1)^2).

Equals Product_{k>=0} (1 + 1/A016754(k)).

A361601 Decimal expansion of the maximum possible disorientation angle between two identical cubes (in radians).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Mar 17 2023

Mackenzie and Thomson (1957) attributed the idea of finding this angle to the British theoretical physicist Frederick Charles Frank (1911-1988), who proposed this problem in 1949.
The disorientation angle between two identical bodies is the least angle of rotation about an axis through the center of mass of one of the bodies that is needed to bring it into the same orientation as the other body. For two cubes with indistinguishable faces, there are 24 rotations angles that will bring the first cube into coincidence with the second, and the disorientation angle is the least of them.
The rotation which achieves this maximum disorientation can be described as a rotation by 90 degrees about any axis parallel to a face diagonal of the cube.
The angle in degrees is 62.7994296198...
The solution to the analogous two-dimensional problem with two squares is the trivial value Pi/4 (A003881).

			1.09605681524062548906172656564125735695942472731840...

D. C. Handscomb, On the random disorientation of two cubes, Canadian Journal of Mathematics, Vol. 10 (1958), pp. 85-88.
J. K. Mackenzie, Second Paper on Statistics Associated with the Random Disorientation of Cubes, Biometrika, Vol. 45, No. 1-2 (1958), pp. 229-240.
J. K. Mackenzie and M. J. Thomson, Some Statistics Associated with the Random Disorientation of Cubes, Biometrika, Vol. 44, No. 1-2 (1957), pp. 205-210.
Wikipedia, Misorientation.
Index entries for transcendental numbers.

Cf. A003881, A201488, A361602, A361603, A361604.

RealDigits[ArcCos[Sqrt[2]/2 - 1/4], 10, 100][[1]]

PARI
```
acos(sqrt(2)/2 - 1/4)
```

Equals arccos(sqrt(2)/2 - 1/4).
Equals 2 * arccos(1/2 + sqrt(2)/4).
Equals 2 * arctan((sqrt(2)-1) * sqrt(5-2*sqrt(2))).

cp's OEIS Frontend

A216546 Decimal expansion of Sum_{k=1..5000000} (-1)^(k-1)/(2k-1).

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A258814 Decimal expansion of the Dirichlet beta function of 7.

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A258816 Decimal expansion of the Dirichlet beta function of 9.

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A084254 Decimal expansion of Sum_{k>=1} 1/(k*(exp(2*Pi*k)-1)).

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A258815 Decimal expansion of the Dirichlet beta function of 8.

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A014200 Number of solutions to x^2 + y^2 <= n, excluding (0,0), divided by 4.

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A177870 Decimal expansion of 3*Pi/4.

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A084254 Decimal expansion of Sum_{k>=1} 1/(k(exp(2Pi*k)-1)).