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Observations:

For all n in this sequence to n = 24, then y = Lambda_n/n follows form: y = (x^2 + x^k) - (floor[z^2/4]) or y = (x^2 + x^k) + (floor[z^2/4]); k = 1 or 2 and z = 0, 1, 3, 6 or 7. y (= A222786) gives the average number of spheres/dimension of the laminated lattice Kissing numbers in A222785.

e.g. Where T_x is the x-th triangular number = (1/2*(x^2 + x)), 2*T_x is the x-th pronic number = (x^2 + x) = floor[(2*x + 1)^2/4], and S_x is the x-th square = (x^2) = floor[(2*x)^2/4]:

For k = 1, z = 0 or 1, then n = {1, 4, 6, 8, 15, 20, 24}, x = {1, 2, 3, 5, 12, 29, 90}, and y = 2*T_x = {2, 6, 12, 30, 156, 870, 8190}.

For k = 2, z = 0 or 1, then n = {1, 5, 7, 23}, x = {1, 2, 3, 45}, and y = 2*T_x + 2*T_(-x) = 2*S_x = {2, 8, 18, 4050}.

For k = 1, z = 3, then n = {3, 7, 12, 16}, x = {2, 4, 7, 16}, and y = 2*T_x - 2*T_1 = {4, 18, 54, 270}.

For k = 1, z = 6, then n = {2, 18}, x = {3, 20}, and y = 2*T_x - S_3 = {3, 411}.

For k = 1, z = 7, then n = {5, 7, 8, 21}, x = {4, 5, 6, 36}, and y = 2*T_x - 2*T_3 = {8, 18, 30, 1320}.

For k = 1, z = 7, then n = {6, 7, 12, 22}, x = {0, 2, 6, 47}, and y = 2*T_x + 2*T_3 = {12, 18, 54, 2268}.

For the special case where k = 1 and z = 0 or 1, then all associated x values follow form (A216162(n) - A216162(n - 2)) [type 1] or (A216162(n) - A216162(n - 1)) [type II] for some n in N. Type II x values = {1, 2, 5, 90} (= A215797(n+1)) are associated with the positive Ramanujan-Nagell triangular numbers = {1, 3, 15, 4095} (= A076046(n+1)) by the formula 1/2*(x^2 + x) = T_x.

Average number of spheres/dimension are given in A222786.

Associated dimensions are given in A211202.

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

Given on p. 8 of Dixon, with "coincidence" involving Fibonacci numbers.

Since there is no indication of how the sequence 1,2,8,24 might be extended, I have marked this as "fini" and "full". - N. J. A. Sloane, Nov 12 2010

Let x = {1, 2, 8, 24}. Then (Lambda_x/x + 1)^2 - 1 = {8, 15, 960, 67092480} and is either a cake number (A000125) or the product of consecutive cake numbers. For instance, 960 = 1 * 2 * 4 * 8 * 15 = (Lambda_8/8 + 1)^2 - 1 and 67092480 = 1 * 2 * 4 * 8 * 15 * 26 * 42 * 64 = (Lambda_24/24 + 1)^2 - 1. This is interesting, at least in part, since x^2 = {1, 4, 64, 576} is also a cake number. - Raphie Frank, Dec 06 2012

Theta series is an element of the space of modular forms on Gamma_1(32) with Kronecker character 8 in modulus 32, weight 31/2, and dimension 62 over the integers.

As of version 2.26-4, the largest rank of a laminated lattice which is recognized by Magma is 31, but laminated lattices of larger rank exist (see Conway and Sloane reference).