Let $a$ be the number of $2$'s. Find the hocus-pocus sum in terms of $a$.

Let $a$ be the number of 2's in the sequences. So, the number of -1's in the sequence is $100-a$.

Now, let $H_A$ be the $hocus-pocus$ sum of the sequence $A$. So,

\[H_A=4\binom{a}{2}+\binom{100-a}{2}-2a(100-a)\]

\[=2a^2-2a+\frac{(100-a)(100-a-1)}{2}-200a+2a^2\]

\[=4.5a^2-301.5a+4950\]

The value of a quadratic equation $ax^2+bx+c$ is the lowest (or highest if $a$ is negative) when the variable is at the vertex, i.e. $a=\frac{-b}{2a}$

The vertex of this equation is $\frac{-(-301.5)}{2\cdot 4.5}=33.5$

But, $a$ must be an integer. So, taking the integers closest to the vertex, taking $a=33$, we see $H_A=-99$

Taking $a=34$, again, $H_A=-99$

So, the least value of $H_A$ is $-99$.