Let p be prime, q = p m , and q − 1 = 7 s . We completely describe the permutation behavior of the binomial P ( x ) = x r ( 1 + x e s ) ( 1 ≤ e ≤ 6 ) over a finite field F q in terms of the sequence { a n } defined by the recurrence relation a n = a n − 1 + 2 a n − 2 − a n − 3 ( n ≥ 3 ) with initial values a 0 = 3 , a 1 = 1 , and a 2 = 5 .