Fibonacci sequence and similar integer sequences are used in many fields from engineering to arts besides basic sciences and mathematics. For every  let  be the number of subsets of           S of the set  including no two consecutive odd integers. We give the generating function and the closed formula of the sequence  obtaining fourth order linear homogeneous recurrence relation with constant coefficients of the integer sequence. The sequence is associated with the Fibonacci sequence and the combinatorial representation of  is obtained.

Fibonacci numbers, consecutive odd integers, generating functions, combinatorial representation.