A019824 -id:A019824 - cp's OEIS frontend

A274014 Decimal expansion of the arc length of an ellipse with semi-major axis 1 and eccentricity sin(Pi/12), an arc length which evaluates without using elliptic integrals (a computation due to Ramanujan).

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

Offset: 1

Jean-François Alcover, Jun 10 2016

			6.176601987658693464745684084107374417575372343469612510291144192254...

Richard E. Crandall, Projects in Scientific Computation, Springer, 1994; see p. 48.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A019824.

p = Sqrt[Pi/Sqrt[3]]*((1 + 1/Sqrt[3])*Gamma[1/3]/Gamma[5/6] + 2*Gamma[5/6]/ Gamma[1/3]);
RealDigits[p, 10, 103][[1]]

sqrt(Pi/sqrt(3))*((1 + 1/sqrt(3))*gamma(1/3)/gamma(5/6) + 2*gamma(5/6)/gamma(1/3)) \\ _G. C. Greubel, Jun 05 2017

Equals (2*((6 + sqrt(3) + 4*sqrt(2 + sqrt(3)))*E((-2 + sqrt(2 + sqrt(3)))^2/(2 + sqrt(2 + sqrt(3)))^2) - 4*sqrt(2 + sqrt(3))*K((-2 + sqrt(2 + sqrt(3)))^2/ (2 + sqrt(2 + sqrt(3)))^2)))/(2 + sqrt(2 + sqrt(3))), where K and E are the elliptic integrals of first and second kind.

Equals sqrt(Pi/sqrt(3))*(((1 + 1/sqrt(3))*Gamma(1/3))/Gamma(5/6) + (2*Gamma(5/6))/Gamma(1/3)).

A343055 Decimal expansion of the imaginary part of i^(1/16), or sin(Pi/32).

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Apr 04 2021

An algebraic number of degree 16 and denominator 2. - Charles R Greathouse IV, Jan 09 2022

			0.09801714032956060199419...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Michael Penn, The exact value of sin 5⅝°??, YouTube video, 2021.
Wikipedia, Trigonometric constants expressed in real radicals
Index entries for algebraic numbers, degree 16

sin(Pi/m): A010527 (m=3), A010503 (m=4), A019845 (m=5), A323601 (m=7), A182168 (m=8), A019829 (m=9), A019827 (m=10), A019824 (m=12), A232736 (m=14), A019821 (m=15), A232738 (m=16), A241243 (m=17), A019819 (m=18), A019818 (m=20), A343054 (m=24), A019815 (m=30), this sequence (m=32), A019814 (m=36).

RealDigits[Sin[Pi/32], 10, 100, -1][[1]] (* Amiram Eldar, Apr 27 2021 *)

PARI
```
imag(I^(1/16))
```
PARI
```
sin(Pi/32)
```
PARI
```
sqrt(2-sqrt(2+sqrt(2+sqrt(2))))/2
```

numerical_approx(sin(pi/32), digits=123) # G. C. Greubel, Sep 30 2022

Equals (1/2) * sqrt(2-sqrt(2+sqrt(2+sqrt(2)))).

One of the 16 real roots of -128*x^2 +2688*x^4 -21504*x^6 +84480*x^8 +32768*x^16 -131072*x^14 +212992*x^12 -180224*x^10 +1 =0. - R. J. Mathar, Aug 29 2025

cp's OEIS Frontend

A274014 Decimal expansion of the arc length of an ellipse with semi-major axis 1 and eccentricity sin(Pi/12), an arc length which evaluates without using elliptic integrals (a computation due to Ramanujan).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A343055 Decimal expansion of the imaginary part of i^(1/16), or sin(Pi/32).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Sage

Formula