A125846 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

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.

MATHEMATICA

PadRight[{1, 1, 3, 1, 4, 24, 63, 288}, 100, 1] (* Paolo Xausa, May 21 2026 *)

CROSSREFS