cp's OEIS Frontend

This sequence is concerned with lattice packings. For unrestricted packings the values are presently known only in dimensions 1, 2, 3, 4, 8 and 24: 2, 6, 12, 24, 240, 196560 (cf. A257479). See Conway and Sloane for details.

a(9) = 272 was determined by Watson (1971). a(10) is probably 336.

Lower bounds for the next 4 terms are 336, 438, 756, 918.

From Natalia L. Skirrow, Jun 04 2023: (Start)

Trivial upper bounds given by A257479 are 553, 869, 1356, 2066.

a(n) coincides with A257479(n) when a lattice achieves the non-lattice-constrained kissing number, for a(0)=0, a(1)=2, a(2)=6 (A_2), a(3)=12 (A_3), a(4)=24 (D_4), a(8)=240 (E_8) and a(24)=196560 (Leech). A002336(n) agrees with a(n) for all n<=9 (and equality is unknown thereafter), and A028923(n)=a(n) iff n<=6. (End)

Two additional terms are known: a(8) = 240, a(24) = 196560 [Odlyzko and Sloane; Levenshtein].

Lower bounds for a(5) onwards are 40, 72, 126, 240 (exact), 306, 500, ... - N. J. A. Sloane, May 15 2015

It seems that, while n is even, a lower bound for a(n) can be written as f(n) = (2n + 2^n) - k(n)^2, where k(2) = 2, for n > 2, k(n) = 2^(n/2) - q, q = {2^t, 3*2^t}, t is an integer, t > 0, 2 <= q <= n, and f(n) <= a(n) (Note: for n <= 24, q = n at n = {4, 6, 8, 24}, q = 0 at n = 2). - Sergey Pavlov, Mar 17 2017

It also seems that, while n is even, an upper bound for a(n) can be written as f(n) = (2n + 2^n) - k(n)^2, where k(2) = 0, for n > 2, k(n) = 2^(n/2) - q, q = {2^t, 3*2^t}, t is an integer, t > 0, n <= q <= 2n, and f(n) >= a(n) (Note: for n <= 24, q = n at n = {2, 4, 12, 16}). - Sergey Pavlov, Mar 19 2017