A125846 Numerator of volume of best symplectic packing of n balls in 4-dimensional ball.

1, 1, 3, 1, 4, 24, 63, 288, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset

1,3

Comments

Explanation, figure, table, references in Traynor. McDuff and Polterovich's existence proof of these packings in nonexplicit; they result from the symplectic blow-up operation. Explicit constructions for n = 8 and n = 9 are still unknown. Biran showed that A125846(n) = A125847(n) = 1 for all n >= 9.

References

P. Biran, Symplectic packing in dimension 4, Geom. Funct. Anal. 7 (1997), pp. 420-437, MR1466333.

D. McDuff and L. Polterovich, Symplectic packings and algebraic geometry, Invent. Math. 115 (1994), pp. 403-434, MR1262938.

Lisa Traynor, Book review (of Embedding problems in symplectic geometry, by Felix Schlenk, deGruyter Expositions in Mathematics, vol. 40, Berlin, 2005), Bull. Amer. Math. Soc. 43 (2006), pp. 593-597.

Links

Table of n, a(n) for n=1..92.

Felix Schlenk, Dusa McDuff and symplectic geometry, Nov 18, 2020.

Index entries for linear recurrences with constant coefficients, signature (1).

Formula

A125846(n)/A125847(n) is maximal symplectic packing density with n balls, as calculated by McDuff and Polterovich.

G.f.: x*(1 + 2*x^2 - 2*x^3 + 3*x^4 + 20*x^5 + 39*x^6 + 225*x^7 - 287*x^8)/(1 - x). - Elmo R. Oliveira, Aug 04 2024

Example

For n = 1..9, the densities are 1, 1/2, 3/4, 1, 4/5, 24/25, 63/64, 288/289, 1.

Crossrefs

Cf. A125847. See A030042/A030043 for an unreduced version.

Sequence in context: A272473 A375299 A362270 * A318382 A340190 A143677

Adjacent sequences: A125843 A125844 A125845 * A125847 A125848 A125849

Keyword

nonn,frac,easy

Author

Jonathan Vos Post, Dec 11 2006

Extensions

Edited by N. J. A. Sloane, Feb 12 2021

Status

approved