  We consider a general family of Carleson sequences associated with dyadic
$A_2$ weights and find sharp -- or, in one case, simply best known -- upper and
lower bounds for their Carleson norms in terms of the $A_2$-characteristic of
the weight. The results obtained make precise and significantly generalize
earlier estimates by Wittwer, Vasyunin, Beznosova, and others. We also record
several corollaries, one of which is a range of new characterizations of dyadic
$A_2.$ Particular emphasis is placed on the relationship between sharp
constants and optimizing sequences of weights; in most cases explicit
optimizers are constructed. Our main estimates arise as consequences of the
exact expressions, or explicit bounds, for the Bellman functions for the
problem, and the paper contains a measure of Bellman-function innovation.
