Evidently this rule gives 0 one ancestor, 1 three. Their squares are 1 and 9, visible respectively as the number of nonzero elements in the individual tensor products. The element sum of N is 10; its eigenvalues are , , contrasted to eigenvalues 0, 1 for and for .
The configurations of length n will have ancestors for an average of 4 each, while the sum of the squares of the number of ancestors will eventually grow according to . The growth could be as small as a factor of 2 per cell or as large as a factor of 4, according to whether all configurations have an equal number of ancestors, or all ancestors map into a single configuration.
The quiescent state will have asymptotically ancestors (whose square is ), an increasingly negligible proportion.
Because the mean is always constant and small, the variance, , will grow asymptotically at half (square root) the rate of the second moment, or by , or 18% per additional cell.
Harold V. McIntosh