Some subsequences of the generalized Fibonacci and Lucas sequences (Q2811707)

From the text: The authors derive first-order nonlinear homogeneous recurrence relations for the subsequence \(\{U_{k^n}\}\) of the generalized Fibonacci \(\{U_n\}\) for odd \(k\), and first-order nonlinear homogeneous recurrence relations for the subsequence \(\{V_{k^n}\}\) of the generalized Lucas sequence \(\{V_n\}\) for both odd and even \(k\). They also give a polynomial representation for the generalized Lucas number \(V_{k^n}\) in terms of generalized Fibonacci numbers \(U_{k^n}\) of degree \(k\) for even \(k\).