2, 8, 18, 4050 follow form (x^2 + x^2), twice a square.
2, 6, 12, 30, 156, 870, 8190 follow form (x^2 + x^1), twice a triangular number.
The first 17 terms in this sequence, as outlined in A211202, follow form (x^2 + x^k) - floor[z^2/4] or (x^2 + x^k) + floor[z^2/4].
OBSERVATION:
There is a curious relationship/coincidence between the values in this sequence equal to either twice a triangular number or twice a square and sequence A266506.
Let p = 13, associated with the 24-dimensional laminated lattice kissing number, 196560, by the simple formula 196560 = (2^p - 2)*(2*p - 2) (see A215929) or, alternatively, (((p^2 + p)/2)^2 - ((p^2 + p)/2))*(2*p - 2).
Let {q} = 2*floor[(n^2 + n)/p] + 1= A011866(n+1) for 0 <= n <= p = {1, 1, 1, 1, 3, 5, 7, 9, 11, 13, 17, 21, 25, 29}.
Let {r} = (A266506(q) - A266506(q -2)) = {-4, -4, -4, -4, 2, 2, 2, 0, 6, 2, 4, 10, 24, 58}. E.g., a(29) - a(27) = 437 - 379 = 58 = r_13.
Let {s} = A266506(r - 2*floor(r/8)), for A266506(r - 2*floor(r/8))<= (p^2 + p); {s} = {4, 4, 4, 4, 2, 2, 2, 2, 3, 2, 1, 4, 46}. Note that all values in {s} also appear in A266504.
Let {t} = A266506(r+1), for A266506(r + 1)<= (p^2 + p); {t} = {5, 5, 5, 5, 1, 1, 1, -1, 5, 1, 3, 11, 181}. Note that all values in {t} also appear in A266505.
Let {r'} = ((r-0)/2)^2 + ((r-0)/2)^1 = {2, 2, 2, 2, 2, 2, 2, 0, 12, 2, 6, 30, 156, 870}.
Let {s'} = ((s-1)/1)^2 + ((s-1)/1)^2 = {18, 18, 18, 18, 2, 2, 2, 2, 8, 2, 0, 18, 4050}.
Let {t'} = ((t-1)/2)^2 + ((t-1)/2)^1 = {6, 6, 6, 6, 0, 0, 0, 0, 6, 0, 2, 30, 8190}.
All terms in {r/2} occur in A002965, bearing in mind that A002965(-4) = -2. All terms in {t} are indices of Ramanujan-Nagell squares (see A060728, A038198 and A227078). All terms in {r'} and {t'} are twice triangular numbers (aka "pronic" numbers) and jointly give the complete set of pronic numbers (n^2 + n) equal to the average number of spheres/dimension for a laminated lattice kissing number in A002336 {0, 2, 6, 12, 30, 156, 870, 8190} associated with dimensions in {0, 1, 4, 6, 8, 15, 20, 24}. Also see "Type I" and "Type II" comments in A216162. All terms in {s'} are twice a square and give the complete set of twice squares (n^2 + n^2) equal to the average number of spheres/dimension for a laminated lattice kissing number in A002336 {0, 2, 8, 18, 4050} associated with dimensions in {0, 1, 5, 7, 23}.