Theorem: Let A(n) be the number of ways to tile, with dominoes, the region in the plane obtained by removing from the c[1] + n + c[2], by , d[1] + n + d[2], rectangle the central , n, by , n, square. where , c[1] = 2, c[2] = 2, d[1] = 2, d[2] = 2 Then 2 A(n) = B(n) where the sequence B(n) is given by the generating function infinity ----- 2 \ n 2 (-3 - t + t ) ) B(n) t = - ---------------------- / 2 ----- (t + 1) (t - 3 t + 1) n = 0 and in Maple input format: -2*(-3-t+t^2)/(t+1)/(t^2-3*t+1) Equivalently, B(n) satisfies the linear recurrence B(n) = 2 B(n - 1) + 2 B(n - 2) - B(n - 3) subject to the initial conditions B(0) = 6, B(1) = 14, B(2) = 38 Proof: Routine. Both sides are C-finite! QED. n B(n) is asymptotic to, 5.483281521 2.618033989 Hence A(n), the number of domino tilings is asymptotic to n 30.06637624 6.854101968 For the sake of Sloane here are the first 31 terms of B(n) starting at n=0. [6, 14, 38, 98, 258, 674, 1766, 4622, 12102, 31682, 82946, 217154, 568518, 1488398, 3896678, 10201634, 26708226, 69923042, 183060902, 479259662, 1254718086, 3284894594, 8599965698, 22515002498, 58945041798, 154320122894, 404015326886, 1057725857762, 2769162246402, 7249760881442] The first 31 terms of A(n), starting at n=0 are [36, 196, 1444, 9604, 66564, 454276, 3118756, 21362884, 146458404, 1003749124, 6880038916, 47155859716, 323212716324, 2215328606404, 15184099435684, 104073336269956, 713329336067076, 4889231802533764, 33511293841053604, 229689823620354244, 1574317475335503396, 10790532493690424836, 73959410006776627204, 506925337484946240004, 3474517952567967072804, 23814700330019262935236, 163228384358801434456996, 1118783990178358655648644, 7668259546898170961945604, 52559032838086684779999364] This took, 0.287, seconds.