Proportion of 1s in a Hadamard matrix

The first post in the recent series of posts on Hadamard matrices describes a way of constructing new Hadamard matrices from two other Hadamard matrices by taking their Kronecker product. Starting with a Hadamard matrix H0 and a Hadamard matrix G, you can construct a sequence of Hadamard matrices by Hn+1 = G ⊗ Hn for positive integers n. This is known as the generalized Sylvester method. Let pn be the proportion of 1s in Hn and let q be the proportion of 1s in G. Then you can show that the recurrence holds pn+1 = q pn + (1 − q)(1 − pn). You can solve the recurrence to show that limn → ∞ pn = ½ and so as the iterations proceed, the ratio of number of 1s to the number of −1s approaches 1. This doesn’t say anything Hadamard matrices in general, but it does apply to all Hadamard matrices created by repeatedly applying the generalized Sylvester method. If you set G and H equal to the matrix then p0 = q = ¾. Then for n = 1, 2, 3, …, 8 the values of pn are 0.625 0.5625 0.53125 0.515625…