Toeplitz–Hessenberg determinant formulas for the sequence \(F_n-1\)

<p>Let <span class="math inline"><em>F</em><sub><em>n</em></sub></span> denote the <span class="math inline"><em>n</em></span>-th Fibonacci number defined by <span class="math inline"><em>F</em><sub><em>n</em></sub> = <em>F</em><sub><em>n</em> − 1</sub> + <em>F</em><sub><em>n</em> − 2</sub></span> if <span class="math inline"><em>n</em> ≥ 2</span>, with <span class="math inline"><em>F</em><sub>0</sub> = 0</span> and <span class="math inline"><em>F</em><sub>1</sub> = 1</span>. In this paper, we find determinant identities for several Toeplitz–Hessenberg matrices whose nonzero entries are derived from the sequence <span class="math inline"><em>k</em><sub><em>n</em> + <em>m</em></sub></span> for various fixed <span class="math inline"><em>m</em></span>, where <span class="math inline"><em>k</em><sub><em>n</em></sub> = <em>F</em><sub><em>n</em></sub> − 1</span>. These results may be obtained algebraically as special cases of more general formulas involving the Horadam numbers and the generating functions for the associated sequences of determinants. Equivalent multi-sum identities featuring sums of products of <span class="math inline"><em>k</em><sub><em>n</em></sub></span> terms with multinomial coefficients may be given, which follow from Trudi’s formula. Connections are made to several OEIS entries that have arisen previously in other contexts, perhaps most notably the Padovan number sequence. Finally, we provide combinatorial proofs of our identities involving <span class="math inline"><em>k</em><sub><em>n</em></sub></span> by enumerating (or finding the sum of signs of) various classes of tilings containing squares, dominos, trominos and a special type of tile which can be of arbitrary length.</p>