The maximal number of distinct entries of an n X n bisymmetric matrix B_n is A002620(n+1) = ((n+1)/2)^2 if n is odd and = n*(n+2)/4 if n is even. See a comment and example under A002620.
Here the first A002620(n+1) positive numbers are used consecutively, that is B_n[i, j] = (i-1)*n - (i-1)^2 + j for j=i..N-(i-1) and i = 1..ceiling(n/2).
Conjecture: a(2*n-1) = (-1)^(n-1)*A034976(n), n >= 1.
For a(2*n)/3 see A259057(n), n >= 1 (assuming that a(2*n) is always divisible by 3).