  Let $f_n(x_1, x_2, \ldots, x_n)$ denote the algebraic normal form (polynomial
form) of a rotation symmetric Boolean function of degree $d$ in $n \geq d$
variables and let $wt(f_n)$ denote the Hamming weight of this function. Let
$(1, a_2, \ldots, a_d)_n$ denote the function $f_n$ of degree $d$ in $n$
variables generated by the monomial $x_1x_{a_2} \cdots x_{a_d}.$ Such a
function $f_n$ is called {\em monomial rotation symmetric} (MRS). It was proved
in a $2012$ paper that for any MRS $f_n$ with $d=3,$ the sequence of weights
$\{w_k = wt(f_k):~k = 3, 4, \ldots\}$ satisfies a homogeneous linear recursion
with integer coefficients. In this paper it is proved that such recursions
exist for any rotation symmetric function $f_n;$ such a function is generated
by some sum of $t$ monomials of various degrees. The last section of the paper
gives a Mathematica program which explicitly computes the homogeneous linear
recursion for the weights, given any rotation symmetric $f_n.$ The reader who
is only interested in finding some recursions can use the program and not be
concerned with the details of the rather complicated proofs in this paper.
