The pebbling number of Fibonacci cubes

The $n$-th Fibonacci cube ${\mathrm{\Gamma }}_n$ is the subgraph of the hypercube $Q_n$ induced by binary strings with no two consecutive ones. We determine $\pi ({\mathrm{\Gamma }}_n)=2^n$ for $n\le 6$, so the pebbling number of ${\mathrm{\Gamma }}_n$ equals that of the ambient hypercube $Q_n$ despite ${\mathrm{\Gamma }}_n$ having far fewer vertices. The lower bound is a standard potential argument. For the upper bound, the Weight Function Lemma yields $2^n+1$ – one too many – so we close the gap by exhaustive MILP verification. We conjecture $\pi ({\mathrm{\Gamma }}_n)=2^n$ for all $n$.