In (West, Discrete Math. 157 (1996) 363–374) it was shown using transfer matrices that the number S_n(123; 3214) of permutations avoiding the patterns 123 and 3214 is the Fibonacci number F_2n (as are also S_n(213; 1234) and S_n(213; 4123)). We now find the transfer matrix for S_n(123; r, r − 1,…,2, 1, r − 1), S_n(213; 1,2,…,r,r + 1), and S_n(213;r + 1, 1, 2,…,r), determine its characteristic polynomial in terms of the Chebyshev polynomials, and go on to determine the generating function as a quotient of modified Chebyshev polynomials. This leads to an asymptotic result for each r which collapses to the exact results 2ⁿ when r = 2 and F_2n when r = 3 and to the Catalan number c_n as r → ∞. We observe that our generating function also enumerates certain lattice paths, plane trees, and directed animals, giving hope that these areas of combinatorics can be applied to enumerating permutations with excluded subsequences.