This is a simple example of an Umbral Scheme for for generating functions for enumerating words in {1,2,3} that avoid 3 consecutive 1's, and any factor of the form 12^(a+1)1^(a+b+1)3^(b+1)1 , for all a,b>=0 The Umbral Scheme is 5 4 t F[1](1, 1) t [F[1](x[1], x[2]) = - --------------------------- - --------------------------- 2 2 2 2 (1 - t x[1]) (1 - t x[2]) (t x[1] - 1) (t x[2] - 1) 4 F[2]() t - ---------------------------, 2 2 (t x[1] - 1) (t x[2] - 1) 3 2 F[2]() = -t - F[1](1, 1) t - F[2]() t (1 + t)], {1, 2} Using this Umbral Scheme, the first 30 terms of the generating function are [1, 3, 9, 26, 76, 221, 642, 1865, 5416, 15728, 45673, 132630, 385145, 1118424, 3247796, 9431289, 27387562, 79530865, 230950040, 670656872, 1947523545, 5655422494, 16422807145, 47690264488, 138487976316, 402155865353, 1167822249554, 3391244350881, 9847849920456, 28597216249136, 83043586549897 ] The whole thing took, 8.900, minutes of CPU time