Consider 8 × n lite arrays of bits. For n=1 there are already 55 (= F₈) such arrays. Let c_8nk be the number of lite 8 × n arrays whose right 8 bits form the binary representation of k. We can now compute the 256 c_8(n+1)k’s from the 256 c_8nk’s. Of these 256 values all but 55 are zero since the right edge, whose bits constitute k, must itself be lite. Each c_8(n+1)k is the sum of the c_8nm’s for which the bits if k and m are disjoint.

Here is code that computes exactly up to n = 12. It reports x = 397743903151933881 12 by 8 lite arrays. log(x)/log(2^12*8) = .6090 .

This code finds exactly x = 18396766424410124752958806046933947217821482942 such 16 by 16 configurations. log(x)/log(2^16*16) = .6003 . Here is its report of the F₁₆ = 2584 counts.

These two calculations include a substantial edge effect since these arrays are embedded in a field of zeros which allows a larger density of 1’s near the edge than inside. Here are some comments on this exact calculation.

We now have information to easily compute the distribution of patterns of bits on a line thru the interior of a randomly selected lite field, or at least the mid-line thru a randomly selected 31 by 16 lite array. A 31 by 16 array can be viewed as two 16 by 16 array sharing one line of 16 bits, the mid-line, which forms an edge to each. Let N_k (= c_16,16,k) be the number of lite 16 by 16 arrays whose left edge is k. (We have just calculated N_k for each k.) There are just ∑_k N_k² 16 × 31 lite arrays for each such has been counted in just one of these summands. The probability of k as the mid-line thru a random 16 × 31 lite array is proportional to N_k²

To select an array at random