A395394 Decimal expansion of the volume of the solid of revolution formed by rotating a unit half-width lemniscate of Bernoulli about the vertical axis.

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

Offset

1,2

Comments

The lemniscate of Bernoulli can be described by (x^2 + y^2)^2 = a^2*(x^2 - y^2) in Cartesian coordinates, or r^2 = a^2*cos(2*phi) in polar coordinates, where a is the half-width. Here we take a = 1.

References

Babu Ram, Engineering Mathematics, Pearson, 2009. See example 8.15.

Links

Table of n, a(n) for n=1..105.

Mandeep Beniwal, Volume of solid of revolution of lemniscate, Smart study helper, YouTube video, 2021.

Amiram Eldar, Illustration of the solid of revolution.

Eric Weisstein's World of Mathematics, Lemniscate.

Eric Weisstein's World of Mathematics, Solid of Revolution.

Wikipedia, Lemniscate of Bernoulli.

Wikipedia, Solid of revolution.

Index entries for transcendental numbers.

Formula

Equals Pi^2/(4*sqrt(2)).

Equals 2 * A328895.

Equals Pi * A193887.

Example

1.7447160499097198835393902340423513224799665677370105...

Mathematica

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

Prog

(PARI) Pi^2/(4*sqrt(2))

Crossrefs

Cf. A091476, A193887, A328895, A343392, A395392, A395393, A395395.

Sequence in context: A377010 A395847 A153349 * A210463 A154172 A021577

Adjacent sequences: A395391 A395392 A395393 * A395395 A395396 A395397

Keyword

nonn,cons,changed

Author

Amiram Eldar, Apr 21 2026

Status

approved