A259830 -id:A259830 - cp's OEIS frontend

A196530 Decimal expansion of log(2+sqrt(3))/sqrt(3).

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

Offset: 0

R. J. Mathar, Oct 03 2011

Equals the value of the Dirichlet L-series of a non-principal character modulo 12 (A110161) at s=1.

			0.7603459963009463475310942548...

L. B. W. Jolley, Summation of series, Dover (1961), eq. (83), page 16.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Étienne Fouvry, Claude Levesque, and Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017.
E. D. Krupnikov, K. S. Kolbig, Some special cases of the generalized hypergeometric function (q+1)Fq, J. Comp. Appl. Math. 78 (1997) 79-95
R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015, Table in section 2.2, L(m=12,r=4,s=1).
Index entries for transcendental numbers.

Cf. A065918, A002194, A110161.

RealDigits[Log[2 + Sqrt[3]]/Sqrt[3], 10, 89][[1]] (* Bruno Berselli, Dec 20 2011 *)

log(sqrt(3)+2)/sqrt(3) \\ Charles R Greathouse IV, May 15 2019

Equals A065918/A002194.
Equals Sum_{n>=1} A110161(n)/n.
Equals Sum_{k>=1} (-1)^(k+1)*2^k/(k * binomial(2*k,k)). - Amiram Eldar, Aug 19 2020
Equals 1/Product_{p prime} (1 - Kronecker(12,p)/p), where Kronecker(12,p) = 0 if p = 2 or 3, 1 if p == 1 or 11 (mod 12) or -1 if p == 5 or 7 (mod 12). - Amiram Eldar, Dec 17 2023
Equals A259830 - 2. - Hugo Pfoertner, Apr 06 2024
Equals (1/2)*2F1(1/2,1;3/2;3/4) [Krupnikov] - R. J. Mathar, Jun 11 2024

A336266 Decimal expansion of (3/16)*Pi.

Original entry on oeis.org

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

Offset: 0

Bernard Schott, Jul 15 2020

Area of one egg of the "double egg" whose polar equation is r(t) = a * cos(t)^2 and a Cartesian equation is (x^2+y^2)^3 = a^2*x^4 is equal to (3/16)*Pi * a^2. See the curve at the Mathcurve link.

			0.58904862254808623221174563436490679078696926...

L. Lorentzen and H. Waadeland, Continued Fractions with Applications, North-Holland 1992, p. 586.

Peter Bala, A note on A336266
Robert Ferréol, Double egg, Mathcurve.
Index to sequences related to curves.
Index entries for transcendental numbers

Cf. A259830 (length of an egg).

Cf. A019692 (2*Pi for deltoid), A122952 (3*Pi for cycloid and nephroid), A180434 (2-Pi/2 for Newton strophoid), A197723 (3*Pi/2 for cardioid), A336308.

Maple
```
evalf(3*Pi/16,140);
```

RealDigits[3*Pi/16, 10, 100][[1]] (* Amiram Eldar, Jul 15 2020 *)

PARI
```
3*Pi/16 \\ Michel Marcus, Jul 15 2020
```

Equals Integral_{t=0..Pi} (1/2) * cos(t)^4 * dt.
Equals Integral_{x=0..oo} 1/(x^2 + 1)^3 dx. - Amiram Eldar, Aug 13 2020
From Peter Bala, Mar 21 2024: (Start)
Equals 1/2 + Sum_{n >= 0} (-1)^n/(u(n)*u(-n)), where the polynomial u(n) = (2*n - 1)*(4*n^2 - 4*n + 3)/3 = A057813(n-1) has its zeros on the vertical line Re(z) = 1/2 in the complex plane. Cf. A336308.
Equals 1/2 + 1/(11 + 3/(12 + 15/(12 + 35/(12 + ... + (4*n^2 - 1)/(12 + ... ))))). See Lorentzen and Waadeland, p. 586, equation 4.7.10 with n = 2. (End)

A376642 Decimal expansion of the area of Moss's egg constructed from a unit-hypotenuse right isosceles triangle.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Sep 30 2024

Moss's egg is an oval named by Dixon (1987) after Stephanie Moss. It is formed by four circular arcs. The shape is composed of the area of a half disk of radius 1/2, circular sector with radius 1-sqrt(2)/2 and central angle Pi/2, and two partially overlapping circular sectors with radius 1 and central angle Pi/4, whose common area is of the unit-hypotenuse right isosceles triangle.

The perimeter of the shape is (3-sqrt(2)/2)*Pi/2.

			0.99547375565275336709301228994445373849422162718740...

Robert Dixon, Mathographics, New York: Dover, 1987. See p. 5.
Anna Weltman, Not Your Average Maths Book, Wide Eyed Editions, 2022. See p. 43.

Michael Borcherds and Bill Lombard, Moss Egg by Robert Dixon, GeoGebra.
Eric Weisstein's World of Mathematics, Moss's Egg.
Wikipedia, Moss's egg.

Similar constants: A093731, A259830, A336266, A336308.

RealDigits[((3 - Sqrt[2])*Pi - 1)/4, 10, 120][[1]]

PARI
```
((3-quadgen(8))*Pi - 1)/4
```

Equals ((3-sqrt(2))*Pi - 1)/4.

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A196530 Decimal expansion of log(2+sqrt(3))/sqrt(3).

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A336266 Decimal expansion of (3/16)*Pi.

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A376642 Decimal expansion of the area of Moss's egg constructed from a unit-hypotenuse right isosceles triangle.

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Mathematica

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