Bit strings withoutq-separation

Abstract

LetB(n, q) denote the number of bit strings of lengthn withoutq-separation. In a bit string withoutq-separation no two 1's are separated by exactlyq − 1 bits.B(n, q) is known to be expressible in terms of a product of powers of Fibonacci numbers. Two new and independent proofs are given. The first proof is by combinatorial enumeration, while the second proof is inductive and expressesB(n, q) in terms of a recurrence relation.