The function $f_{}^{}$ has the property that, for each real number $x,\,$

$f(x)+f(x-1) = x^2.\,$
If $f(19)=94,\,$ what is the remainder when $f(94)\,$ is divided by $1000$?
\begin{align*}f(94)&=94^2-f(93)=94^2-93^2+f(92)=94^2-93^2+92^2-f(91)=\cdots \\ &= (94^2-93^2) + (92^2-91^2) +\cdots+ (22^2-21^2)+ 20^2-f(19) \\ &= 94+93+\cdots+21+400-94  \\ &= 4561 \end{align*}
So, the remainder is $\boxed{561}$.