Fibonacci Partitions

Then f(z) is also an analytic function without zeros on compact subsets of the unit disk. We have A * ) = n o * * ) 1 = z v < °=0(4) «>1 «>0 Definition 1: Let r(#)? r^(n), r0(w) denote, respectively, the number of partitions of n into distinct parts, evenly many distinct parts, oddly many distinct parts from {un}. Let r (0) = rE(0) = l, r0(0) = 0. If an =rE(n)-rQ(n)9 then U„ is the number of partitions of n all of whose parts belong to {un}, that is, f(z) is the generating function for {un}. Since f(z)*g(z) = l, w e obtain the recurrence relation: