On the number of combinations without unit separation

Let f(n, k) denote the number of ways of selecting k objects from n objects arrayed in a line with no two selected having unit separation (i.e., having exactly one object between them). Then, if $\text{n ? 2(k ? 1), f(n,k)=}\text{∑}\text{i=0}\text{κ}\text{(}\text{n?k+I?2i}\text{k?2i}\text{)}$ (where $\text{κ =}\text{k}\text{2}\text{]}$). If n < 2(k ? 1), then f(n, k) = 0. In addition, f(n, k) satisfies the recurrence relation f(n, k) = f(n ? 1, k) + f(n ? 3, k ? 1) + f(n ? 4, k ? 2). If the objects are arrayed in a circle, and the corresponding number is denoted by g(n, k), then for n > 3, g(n, k) = f(n ? 2, k) + 2f(n ? 5, k ? 1) + 3f(n ? 6, k ? 2). In particular, if n ? 2k + 1 then $\text{(n,k)=(}\text{n?k}\text{k}\text{)+(}\text{n?k?1}\text{k?1}\text{)}$.