Quadratic-size formula with prescribed model count ($|{\phi }_k|=O(n^2),\#\mathrm{SAT}({\phi }_k)=k$)

For each $0\le k\le 2^n$, a propositional formula over $n$ specified variables can have exactly $k$ satisfying assignments and length $O(n^2)$. For $0<k<2^n$, interpret a Boolean valuation as the binary number $x_1\cdots x_n$ and test whether it is less than $k=b_1\cdots b_n$.

For every position $i$ with $b_i=1$, use a conjunction asserting that all earlier bits equal $b_j$ and $x_i=0$, leaving later bits unrestricted. Take the disjunction of these terms. Each satisfying assignment has a unique first differing position, so the terms are disjoint, and their counts sum to ${\sum }_{i:b_i=1}{2}^{n-i}=k$. Their combined length is $O(1+2+\cdots +n)=O(n^2)$. Use logical falsity and logical truth at the two endpoints. The count is over all $n$ variables, including those absent from an individual term or from the whole formula.