A019670 -id:A019670 - 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

A100044 Decimal expansion of Pi^2/9.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Oct 31 2004

The Dirichlet L-series for the principal character mod 6 (which is A120325 shifted left) evaluated at 2. - R. J. Mathar, Jul 20 2012

Equals the asymptotic mean of the abundancy index of the numbers coprime to 6 (A007310). - Amiram Eldar, May 12 2023

			1.096622711232150957648276777764...

F. Aubonnet, D. Guinin, and B.Joppin, Précis de Mathématiques, Analyse 2, Classes Préparatoires, Premier Cycle Universitaire, Bréal, 1990, Exercice 908, pages 82 and 91-92.
L. B. W. Jolley, Summation of Series, Dover, 1961.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Jean-Paul Allouche and Jeffrey Shallit, Sums of digits and the Hurwitz zeta function, in: K. Nagasaka and E. Fouvry (eds.), Analytic Number Theory, Lecture Notes in Mathematics, Vol. 1434, Springer, Berlin, Heidelberg, 1990, pp. 19-30.
Eugène-Charles Catalan, Mémoire sur la transformation des séries et sur quelques intégrales définies, Mémoires de l'Académie royale de Belgique, 1867, Vol. 33, pp. 1-50.
R. J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, arXiv:1008.2547 [math.NT], 2010-2015, Table 22.
Eric Weisstein's World of Mathematics, Digit Sum.
Index entries for transcendental numbers.

Cf. A000120, A007310, A008833, A019670, A086463, A091999, A104777, A120325, A112093, A112094, A130549, A130550.

RealDigits[Pi^2/9, 10, 110][[1]] (* G. C. Greubel, Feb 17 2017 *)

default(realprecision, 110); Pi^2/9 \\ G. C. Greubel, Feb 17 2017

numerical_approx(pi^2/9, digits=120) # G. C. Greubel, Jun 02 2021

Equals 1 + (1/2)*(1/3)*(1/2) + (1/3)*(1*2)/(3*5)*(1/2)^2 + (1/4) *(1*2*3)/(3*5*7)*(1/2)^3 + .... [Jolley eq 277]
Equals 1/1^2 + 1/5^2 + 1/7^2 + 1/11^2 + 1/13^2 + 1/17^2 + .... - R. J. Mathar, Jul 20 2012
Equals 2*Sum_{n>=1} 1/(6*n*(3*n + (-1)^n - 3) - 3*(-1)^n + 5) = 2*Sum_{n>=1} 1/(2*A104777(n)). - Alexander R. Povolotsky, May 18 2014
Equals A019670^2. - Michel Marcus, May 19 2014
Equals 2*A086463 = 2*Sum_{n>=1} 1/A091999(n)^2, equivalent to the formula of 2012 above. - Alexander R. Povolotsky, May 20 2014
Equals 3F2(1,1,1; 3/2,2 ; 1/4), following from Clausen's formula of J. Reine Angew. Math 3 (1828) for squares of 2F1() as noted in A019670. - R. J. Mathar, Oct 16 2015
Equals Product_{n >= 3} prime(n)^2 / (prime(n)^2 - 1), Euler's prime product, excluding first two primes. - Fred Daniel Kline, Jun 09 2016
Equals Integral_{x=0..oo} log(x)/(x^6 - 1) dx. - Amiram Eldar, Aug 12 2020
Equals Sum_{k>=1} A000120(k) * (2*k+1)/(k^2*(k+1)^2) (Allouche and Shallit, 1990). - Amiram Eldar, Jun 01 2021
Equals Integral_{x=0..1} log(1+x+x^2)/x dx (Aubonnet). - Bernard Schott, Feb 04 2022
Equals Sum_{k>=1} A008833(k)/k^4. - Amiram Eldar, Jan 25 2024
Continued fraction expansion: 1/(1 - 1/(13 - 48/(34 - 270/(65 - ... - 2*(2*n-1)*n^3/((5*n^2+6*n+2) - ... ))))). See A130549. - Peter Bala, Feb 16 2024
Equals Sum_{k >= 0} 1/((k + 1)*(2*k + 1)*binomial(2*k, k)). See Catalan, Section 21, equation 30. - Peter Bala, Aug 14 2024

A035154 a(n) = Sum_{d|n} Kronecker(-36, d).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 0, 1, 1, 2, 0, 1, 2, 0, 2, 1, 2, 1, 0, 2, 0, 0, 0, 1, 3, 2, 1, 0, 2, 2, 0, 1, 0, 2, 0, 1, 2, 0, 2, 2, 2, 0, 0, 0, 2, 0, 0, 1, 1, 3, 2, 2, 2, 1, 0, 0, 0, 2, 0, 2, 2, 0, 0, 1, 4, 0, 0, 2, 0, 0, 0, 1, 2, 2, 3, 0, 0, 2, 0, 2, 1, 2, 0, 0, 4, 0, 2, 0, 2, 2, 0, 0, 0, 0, 0, 1, 2, 1, 0, 3, 2, 2, 0, 2, 0

Offset: 1

N. J. A. Sloane

			G.f. = x + x^2 + x^3 + x^4 + 2*x^5 + x^6 + x^8 + x^9 + 2*x^10 + x^12 + 2*x^13 + ...

Bruce C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, 1994, see p. 197, Entry 44.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A002654, A008441, A019670, A122856, A122857, A122865, A125079.

a[ n_] := If[ n < 1, 0, Sum[ KroneckerSymbol[ -36, d], { d, Divisors[ n]}]]; (* Michael Somos, Jun 24 2011 *)
a[ n_] := SeriesCoefficient[ (-2 + EllipticTheta[ 3, 0, q]^2 + EllipticTheta[ 3, 0, q^3]^2) / 4, {q, 0, n}]; (* Michael Somos, Jul 09 2013 *)

{a(n) = if( n<1, 0, sumdiv(n, d, kronecker( -36, d)))}; /* Michael Somos, Jul 30 2006 */

{a(n) = if( n<1, 0, direuler(p=2, n, 1 / ((1 - X) * (1 - kronecker( -36, p) * X))) [n])}; /* Michael Somos, Jul 30 2006 */

{a(n)=polcoeff(sum(m=0,n\6+1,(-1)^m*(x^(6*m+1)/(1-x^(6*m+1)+x*O(x^n)) + x^(6*m+5)/(1-x^(6*m+5)+x*O(x^n)))),n)} /* Paul D. Hanna */

Expansion of -1 + (theta_3(q)^2 + theta_3(q^3)^2) / 2 in powers of q. - Michael Somos, Jul 09 2013
From Michael Somos, Jul 30 2006: (Start)
Moebius transform is period 12 sequence [1, 0, 0, 0, 1, 0, -1, 0, 0, 0, -1, 0, ...].
Multiplicative with a(2^e) = a(3^e) = 1, a(p^e) = e+1 if p == 1(mod 4), a(p^e) = (1 + (-1)^e) / 2 if p == 3(mod 4). (End)
Dirichlet g.f.: zeta(s) * L(chi,s) where chi(n) = Kronecker( -36, n). Sum_{n>0} a(n) / n^s = Product_{p prime} 1 / ((1 - p^-s) * (1 - Kronecker( -36, p) * p^-s)). - Michael Somos, Jun 24 2011
a(2*n) = a(3*n) = a(n). a(2*n + 1) = A125079(n). a(3*n + 1) = A122865(n). a(3*n + 2) = A122856(n). a(4*n + 1) = A008441(n).
2 * a(n) = A122857(n) unless n=0. - Michael Somos, Jul 09 2013
G.f.: Sum_{n>=0} (-1)^n*( x^(6*n+1)/(1-x^(6*n+1)) + x^(6*n+5)/(1-x^(6*n+5)) ). - Paul D. Hanna, Dec 14 2011
G.f.: x/(1-x) + x^5/(1-x^5) - x^7/(1-x^7) - x^11/(1-x^11) + x^13/(1-x^13) + x^17/(1-x^17) --++ ...
a(n) = A002654(n) + A002654(3*n). - Michael Somos, Jan 25 2017
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/3 = 1.0471975... (A019670). - Amiram Eldar, Nov 17 2023

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

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

A125079 Excess of number of divisors of 2n+1 of form 12k+1, 12k+5 over those of form 12k+7, 12k+11.

Original entry on oeis.org

1, 1, 2, 0, 1, 0, 2, 2, 2, 0, 0, 0, 3, 1, 2, 0, 0, 0, 2, 2, 2, 0, 2, 0, 1, 2, 2, 0, 0, 0, 2, 0, 4, 0, 0, 0, 2, 3, 0, 0, 1, 0, 4, 2, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 2, 2, 0, 2, 0, 1, 2, 4, 0, 0, 0, 0, 2, 2, 0, 0, 0, 4, 1, 2, 0, 2, 0, 2, 2, 0, 0, 0, 0, 3, 0, 2, 0, 0, 0, 2, 2, 4, 0, 0, 0, 2, 4, 2, 0, 0, 0, 4, 0, 0

Offset: 0

Michael Somos, Nov 18 2006

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			1 + x + 2*x^2 + x^4 + 2*x^6 + 2*x^7 + 2*x^8 + 3*x^12 + x^13 + 2*x^14 + ...
q + q^3 + 2*q^5 + q^9 + 2*q^13 + 2*q^15 + 2*q^17 + 3*q^25 + q^27 + 2*q^29 + ...

Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 82, Eq. (32.56).

Antti Karttunen, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A002175, A008441, A019670, A035154, A121444.

Cf. A000122, A000700, A010054, A121373.

a[ n_] := If[ n < 0, 0, With[{m = 2 n + 1}, Sum[ KroneckerSymbol[ -36, d], { d, Divisors[ m]}]]]

{a(n) = if( n<0, 0, n = 2*n+1; sumdiv( n, d, kronecker( -36, d)))}

{a(n) = if( n<0, 0, n = 2*n+1; sumdiv( n, d, kronecker( 6, d) * (-1)^(d\12)))}

{a(n) = if( n<0, 0, if( n%6==1, n\=3, 1); sumdiv( 2*n + 1, d, kronecker( -4, d)) )}

{a(n) = local(A, p, e); if( n<0, 0, n = 2*n + 1; A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, 0, if( p==3, 1, if( p%4==1, e+1, !(e%2)))))))}

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^3 * eta(x^4 + A) * eta(x^12 + A) / (eta(x + A) * eta(x^6 + A)^2), n))}

a008441(n) = sumdiv(4*n+1, d, (-1)^(d\2));
a(n) = if(n%2==0, a008441(n/2), if(n%6==1, a008441((n-1)/6))); \\ Seiichi Manyama, Oct 11 2024

Expansion of q^(-1/2) * eta(q^3)^3 *eta(q^4) * eta(q^12) / (eta(q) * eta(q^6)^2) in powers of q.
Expansion of phi(-q^3) * psi(-q^3) / (chi(-q) * chi(-q^2)) in powers of q where phi(), psi(), chi() are Ramanujan theta functions.
Euler transform of period 12 sequence [ 1, 1, -2, 0, 1, 0, 1, 0, -2, 1, 1, -2, ...].
a(n) = b(2*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(3^e) = 1, b(p^e) = e+1 if p == 1 (mod 4), b(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4).
G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 2 (t/i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A138745.
a(6*n + 3) = a(6*n + 5) = 0. a(6*n) = A002175(n). a(6*n + 1) = a(2*n) = A008441(n). a(6*n + 2) = 2 * A121444(n). a(n) = A035154(2*n + 1).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/3 = 1.0471975... (A019670). - Amiram Eldar, Dec 28 2023
a(6*n + 3) = a(6*n + 5) = 0. a(6*n + 1) = A008441(n). a(6*n + k) = A008441(3*n + k/2) for k=0,2,4. - Seiichi Manyama Oct 11 2024

A019683 Decimal expansion of Pi/16.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

			Pi/16 = 0.19634954084936207740391521145496893026232308746094411381... - _Vladimir Joseph Stephan Orlovsky_, Dec 02 2009

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.4.2, p. 494.

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

Cf. A157332, A000796, A003881, A019669, A019670, A019675, A019679, A244978.

R:= RealField(100); Pi(R)/16; // G. C. Greubel, Aug 26 2019

RealDigits[N[Pi/16,6! ]] (* Vladimir Joseph Stephan Orlovsky, Dec 02 2009 *)

default(realprecision, 100); Pi/16 \\ G. C. Greubel, Aug 26 2019

numerical_approx(pi/16, digits=100) # G. C. Greubel, Aug 26 2019

From  Peter Bala, Oct 27 2019: (Start)
Equals Integral_{x = 0..1} x^2*sqrt(1 - x^2) dx = Integral_{x = 0..1} x^3*sqrt(1 - x^8) dx.
Equals Integral_{x = 0..inf} x^2/(1 + x^2)^3 dx. (End)
From Amiram Eldar, Aug 04 2020: (Start)
Equals Sum_{k>=1} sin(k)^3 * cos(k)/k.
Equals Sum_{k>=1} sin(k)^3 * cos(k)^2/k.
Equals Sum_{k>=1} (-1)^(k+1) * sin((2*k-1)/4)/(2*k-1)^2. (End)

A164109 Decimal expansion of Pi^4/3.

Original entry on oeis.org

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

Offset: 2

R. J. Mathar, Aug 10 2009

Surface area of the 8-dimensional unit sphere.

			32.4696970113341457454801108962350370832425285...

G. C. Greubel, Table of n, a(n) for n = 2..5000
Eric Weisstein's World of Mathematics, Hypersphere.
Wikipedia, Hypersphere.
Ce Xu and Jianqiang Zhao, Sun's Three Conjectures on Apéry-like Sums Involving Harmonic Numbers, arXiv:2203.04184 [math.NT], 2022.
Index entries for transcendental numbers.

Cf. A019692, A019694, A164102, A164104, A091925, A164107.

Cf. A001008, A002805.

RealDigits[Pi^4/3,10,120][[1]] (* Harvey P. Dale, Jul 31 2014 *)

PARI
```
Pi^4/3 \\ G. C. Greubel, Sep 11 2017
```

Equals 8*A164108 = A019670*A091925 = A092425/3.

Pi^4/30 = Sum_{k>=1} H(k)*H(k+1)/(k+1)^2, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number. - Amiram Eldar, Aug 19 2025

A175617 Decimal expansion of product_{n>=2} (1-n^(-6)).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jul 26 2010

			0.9826842777...

E. Weisstein, Infinite Product, Mathworld.

Cf. A109219, A175615, A175619, A144665, A258871.

cosh(Pi*sqrt(3)/2)^2/6/Pi^2 ; evalf(%) ;

RealDigits[(1 + Cosh[Sqrt[3]*Pi])/(12*Pi^2), 10, 105] // First (* Jean-François Alcover, Feb 12 2013 *)

exp(suminf(j=1, (1 - zeta(6*j))/j)) \\ Vaclav Kotesovec, Apr 27 2020

Equals product_{t=1..5} 1/Gamma(2-exp(Pi*i*t/3)), where i is the imaginary unit and Pi/3 = A019670.

Equals exp(Sum_{j>=1} (1 - zeta(6*j))/j). - Vaclav Kotesovec, Apr 27 2020

A352324 Decimal expansion of 4Pi / (5sqrt(10-2*sqrt(5))).

Original entry on oeis.org

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

Offset: 1

Bernard Schott, Mar 12 2022

Cauchy's residue theorem implies that Integral_{x=0..oo} 1/(1 + x^m) dx = (Pi/m) * csc(Pi/m); this is the case m = 5.

The area of a circle circumscribing a unit-area regular decagon.

			1.0689593321155951134251843725068826399014509252665...

Jean-François Pabion, Éléments d'Analyse Complexe, licence de Mathématiques, page 111, Ellipses, 1995.

Index entries for transcendental numbers.

Cf. A019694, A047209, A121570, A371604, A377405.

Integral_{x=0..oo} 1/(1+x^m) dx: A019669 (m=2), A248897 (m=3), A093954 (m=4), this sequence (m=5), A019670 (m=6), A352125 (m=8), A094888 (m=10).

evalf(4*Pi / (5*(sqrt(10-2sqrt(5)))), 100);

First[RealDigits[N[4Pi/(5Sqrt[10-2Sqrt[5]]), 93]]] (* Stefano Spezia, Mar 12 2022 *)

Equals Integral_{x=0..oo} 1/(1 + x^5) dx.
Equals (Pi/5) *csc(Pi/5).
Equals (1/2) * A019694 * A121570.
Equals 1/Product_{k>=1} (1 - 1/(5*k)^2). - Amiram Eldar, Mar 12 2022
Equals Product_{k>=2} (1 + (-1)^k/A047209(k)). - Amiram Eldar, Nov 22 2024
Equals 1/A371604 = A377405/5. - Hugo Pfoertner, Nov 22 2024

cp's OEIS Frontend

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