Fibonacci-like Sequences

Consider the following recurrence relation:$$\begin{gathered} \text{a}(n)=\text{a}(n-1)+\text{a}(n-8) \\ \text{with }\text{a}(i)=1\text{ for }i=0\dots 7 \end{gathered}$$The ratio of successive terms approach the golden ratio $\varphi$ just as the terms in the Fibonacci sequence do. Naturally, the terms in the sequence take a while to grow larger. Here are the initial terms:

1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 14, 18, 23, 29, 36, 44, 53, 64, 78, 96, 119, 148, 184, 228, 281, 345, 423, 519, 638, 786, 970, 1198, 1479, 1824, 2247, 2766, 3404, 4190, 5160, 6358, 7837, 9661, 11908, 14674, 18078, 22268, 27428, 33786, 41623

These terms form OEIS A005710. The generating function (permalink) for this sequence is:$$\frac{1}{1-x-x^8}$$In general, we have:$$\begin{gathered} a(n)=a(n-1)+a(n-m) \\ \text{ with }a(n)=1\text{ for }n=0\dots m-1 \end{gathered}$$The generating function is:$$\frac{1}{1-x-x^m}$$In the case of $m=2$, we get the terms in the Fibonacci sequence.