A computational substantiation of the d-step approach to the number of distinct squares problem

Motivated by the recent validation of the d-step approach for the number of runs problem, we investigate the largest possible number σd(n) of distinct primitively rooted squares over all strings of length n with exactly d distinct symbols. New properties of σd(n) are presented, and the notion of s-cover is introduced with an emphasis on the recursive computational determination of σd(n). In particular, we were able to determine all values of σ2(n) for n≤70, σ3(n) for n≤45 and σ4(n) for n≤38. These computations reveal the unexpected existence of pairs (d,n) satisfying σd+1(n+2)−σd(n)>1 such as (2, 33) and (2, 34), and of three consecutive equal values: σ2(31)=σ2(32)=σ2(33). Noticeably, we show that among all strings of length 33, the maximum number of distinct primitively rooted squares cannot be achieved by a non-ternary string.