Subsets without \(q\)-separation and binomial products of Fibonacci numbers (Q1175959)

Let \(T(n,q)\) be the number of subsets of \(\{1,2,\ldots,n\}\) where no two consecutive integers have difference \(q\). Then \(T(n,1)\) is the Fibonacci number \(F_{n+2}\). By considering the subsets \(S_ i\) of all \(k\equiv i\bmod q\) and writing \(n=mq+r\), \(0\leq r<q\), the authors show that \[ T(n,q)=F^{q-r}_{m+2}F^ r_{m+3} \] and obtain some corollaries.