A093731 -id:A093731 - cp's OEIS frontend

A358981 Decimal expansion of Pi/3 - sqrt(3)/4.

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

Offset: 0

Michal Paulovic, Dec 08 2022

The constant is the area of a circular segment bounded by an arc of 2*Pi/3 radians (120 degrees) of a unit circle and by a chord of length sqrt(3). Three such segments result when an equilateral triangle with side length sqrt(3) is circumscribed by a unit circle. The area of each segment is:
  A = (R^2 / 2) * (theta - sin(theta))
  A = (1^2 / 2) * (2*Pi/3 - sin(2*Pi/3))
  A = (1 / 2) * (2*Pi/3 - sqrt(3)/2)
  A = Pi/3 - sqrt(3)/4 = (Pi - 3*sqrt(3)/4) / 3 = 0.61418484...
where Pi (A000796) is the area of the circle, and 3*sqrt(3)/4 (A104954) is the area of the inscribed equilateral triangle.
The sagitta (height) of the circular segment is:
  h = R * (1 - cos(theta/2))
  h = 1 * (1 - cos(Pi/3))
  h = 1 - 1/2 = 0.5 (A020761)

			0.6141848493043784...

Index entries for transcendental numbers

Cf. A000796, A019670, A020761, A093731, A104954, A120011.

Maple
```
evalf(Pi/3-sqrt(3)/4);
```

RealDigits[Pi/3 - Sqrt[3]/4, 10, 100][[1]]

PARI
```
Pi/3 - sqrt(3)/4
```

Equals A019670 - A120011. - Omar E. Pol, Dec 08 2022

Equals A093731 / 2. - Michal Paulovic, Mar 08 2024

A101418 Floor of the area of a lens constructed using circular arcs of radius n.

Original entry on oeis.org

1, 4, 11, 19, 30, 44, 60, 78, 99, 122, 148, 176, 207, 240, 276, 314, 354, 397, 443, 491, 541, 594, 649, 707, 767, 830, 895, 963, 1033, 1105, 1180, 1257, 1337, 1419, 1504, 1591, 1681, 1773, 1868, 1965, 2064, 2166, 2271, 2378, 2487, 2599, 2713, 2830, 2949

Offset: 1

Joseph Biberstine (jrbibers(AT)indiana.edu), Jan 16 2005

nonn

			a(2) = 4 because a lens given by the intersection of two circles of radius two has an area of approximately 4.91347...

Eric Weisstein's World of Mathematics, Lens

Cf. A093731.

Table[Floor[(4*Pi - 3*Sqrt[3])/6*r^2], {r, 1, 60}]

a(n)=(4*Pi-sqrt(27))*n^2\6 \\ Charles R Greathouse IV, Nov 27 2016

a(n) = floor((4*Pi - 3*sqrt(3))/6*n^2).

A276478 Number of points in square lattice in and on the boundary of the area encompassed by two arcs of radius n and centers at (0,0) and (n,0).

Original entry on oeis.org

1, 2, 5, 12, 19, 34, 45, 56, 77, 98, 127, 148, 169, 206, 239, 280, 311, 350, 393, 440, 495, 534, 593, 644, 697, 770, 827, 896, 957, 1026, 1105, 1168, 1255, 1330, 1417, 1512, 1579, 1678, 1759, 1868, 1969, 2050, 2159, 2256, 2377, 2490, 2585, 2704, 2811, 2942

Offset: 0

Christina Steffan, Sep 05 2016

nonn

Wikipedia, Vesica piscis

Cf. A057961, A093731.

a(n) = my(m = floor(n*sqrt(3)/2)); 1 - 3*n - 2*(n-1)*m + 4*sum(k=0, m, sqrtint(n^2-k^2)); \\ Michel Marcus, Mar 07 2021

a(n) = 1 - 3*n - 2*(n-1)*m(n) + 4 * Sum_{k=0..m(n)} floor(sqrt(n^2-k^2)) where m(n) = floor(n*sqrt(3)/2). - Franz Vrabec, Oct 02 2016

a(n)/n^2 tends to A093731 as n tends to infinity. - Rémy Sigrist, Mar 07 2021

More terms from Franz Vrabec, Oct 02 2016

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.

cp's OEIS Frontend

A358981 Decimal expansion of Pi/3 - sqrt(3)/4.

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A101418 Floor of the area of a lens constructed using circular arcs of radius n.

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A276478 Number of points in square lattice in and on the boundary of the area encompassed by two arcs of radius n and centers at (0,0) and (n,0).

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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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