Quadratic truncated rotation symmetric Boolean functions

A Boolean function g_n in n variables is rotation symmetric (RS) if it is invariant under powers of ρ(x₁,…,x_n)=(x₂,…,x_n,x₁). An RS function is monomial rotation symmetric (MRS) if it is generated by applying powers of ρ to a single monomial, say x₁x_a(2)…x_a(d), where d is the degree of the function. An MRS function in n variables is called truncated rotation symmetric (TRS) if the function stops the expansion for the n-variable MRS function at the term where x_n first occurs. Truncated functions are important because they are used in the computation of linear recursions which the weights of any RS functions are known to satisfy. Computing these recursions in general is very complex. This paper proves that for the quadratic TRS functions, an explicit formula for the generating function for the weights can be proved. This removes the need for the complex computation and makes the weight computation much simpler.