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

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

Author

Bernard Schott, Jul 15 2020

Keywords

• nonn
• cons

Comments

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.

Examples

			0.58904862254808623221174563436490679078696926...
		

References

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

Links

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

Crossrefs

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.

Programs

Maple

evalf(3*Pi/16,140);

Mathematica

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

PARI

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

Formula

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)