On the Lucas matrix of order \(2^k\) sequence \(\{L_n^{(2^k)}\}\). (Q2716017)

Starting with the sequence of Lucas numbers \((L_n)_{n\geq 0}=(2,1,3,4,7,11,\dots )\), extended to \(n<0\) by the recurrence \(L_{n+1}=L_n+L_{n-1}\), the author recursively defines ``Lucas'' matrix \(L_n^{(2^k)}\) of order \(2^k\), \(k\geq 0\), by composing it of the matrices \(L_{n+1}^{(2^{k-1})}\), \(L_n^{(2^{k-1})}\), \(L_n^{(2^{k-1})}\), and \(L_{n-1}^{(2^{k-1})}\) on the positions 1-1, 1-2, 2-1, and 2-2, respectively. Several simple properties of these matrices are proved, such as the reciprocity formula \(L_{-n}^{(2^k)}=(-1)^nEL_n^{(2^k)}E\) where \(E\) has certain pattern of \(1\)s and \(-1\)s on the second diagonal and \(0\)s elsewhere (Theorem 1) or the summation formula \(L_{n+p}^{(2^k)}=\sum _{i=0}^p{\binom {p}{i}}L_{n-i}^{(2^k)}\) (Theorem 8).