--------------------------------------- The Generalized Capsules for the coefficient of q^(m*n+m-1) mod m for the unique formal power series satisying thefunctional equation m q f(q ) f(q) = 1 + ----------------- 2 (1 - q) (-q + 1) for m from 3 to 20, except those that are 2 mod 4 are as follows m is, 3 1 + 2 q [---------------, [0, 1, 1], [1, 1, 2]] 3 2 q - q - q + 1 m is, 4 2 [------------, [0, 1, 3, 3, 1], [2, 2, 0, 2, 0]] 2 q - 2 q + 1 m is, 5 2 + 3 q [---------------, [0, 1, 1, 3, 3, 1, 1], [2, 2, 4, 1, 3, 2, 1]] 3 2 q - q - q + 1 m is, 7 3 + 4 q [---------------, [0, 1, 1, 3, 3, 6, 6, 3, 3, 1, 1], 3 2 q - q - q + 1 [3, 3, 6, 2, 5, 4, 0, 2, 1, 6, 1]] m is, 8 4 [------------, [0, 1, 3, 7, 5, 6, 2, 2, 6, 5, 7, 3, 1], 2 q - 2 q + 1 [4, 4, 0, 4, 0, 0, 4, 0, 4, 0, 4, 0, 4]] m is, 9 4 + 5 q [---------------, [0, 1, 1, 3, 3, 6, 6, 1, 1, 6, 6, 3, 3, 1, 1], 3 2 q - q - q + 1 [4, 4, 8, 3, 7, 6, 1, 4, 8, 6, 5, 7, 1, 7, 5]] m is, 11 5 + 6 q [---------------, [0, 1, 1, 3, 3, 6, 6, 10, 10, 4, 4, 10, 10, 6, 6, 3, 3, 1, 1], 3 2 q - q - q + 1 [5, 5, 10, 4, 9, 8, 2, 6, 0, 9, 3, 6, 5, 2, 6, 8, 6, 2, 5]] m is, 12 6 [------------, 2 q - 2 q + 1 [0, 1, 3, 7, 1, 10, 10, 2, 10, 11, 5, 5, 11, 10, 2, 10, 10, 1, 7, 3, 1], [6, 6, 0, 6, 0, 0, 6, 0, 6, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0, 6, 0]] m is, 13 6 + 7 q [---------------, 3 2 q - q - q + 1 [0, 1, 1, 3, 3, 6, 6, 10, 10, 2, 2, 8, 8, 2, 2, 10, 10, 6, 6, 3, 3, 1, 1], [6, 6, 12, 5, 11, 10, 3, 8, 1, 12, 5, 9, 2, 12, 11, 1, 6, 2, 0, 2, 6, 1, 11 ]] m is, 15 7 + 8 q [---------------, [0, 1, 1, 3, 3, 6, 6, 10, 10, 0, 0, 6, 6, 13, 13, 6, 6, 0, 0, 3 2 q - q - q + 1 10, 10, 6, 6, 3, 3, 1, 1], [7, 7, 14, 6, 13, 12, 4, 10, 2, 0, 7, 12, 4, 1, 8, 12, 11, 7, 13, 1, 14, 9, 14, 1, 13, 7, 11]] m is, 16 8 [------------, [0, 1, 3, 7, 13, 6, 2, 2, 6, 15, 13, 1, 11, 12, 4, 4, 12, 11, 1, 2 q - 2 q + 1 13, 15, 6, 2, 2, 6, 13, 7, 3, 1], [8, 8, 0, 8, 0, 0, 8, 0, 8, 8, 0, 8, 0, 0, 8, 0, 8, 0, 8, 0, 8, 0, 8, 0, 8, 0, 8, 0, 8]] m is, 17 8 + 9 q [---------------, [0, 1, 1, 3, 3, 6, 6, 10, 10, 15, 15, 4, 4, 11, 11, 2, 2, 11, 3 2 q - q - q + 1 11, 4, 4, 15, 15, 10, 10, 6, 6, 3, 3, 1, 1], [8, 8, 16, 7, 15, 14, 5, 12, 3, 1, 9, 15, 6, 3, 11, 16, 7, 3, 2, 6, 13, 8, 6, 9, 15, 9, 6, 8, 13, 6, 2]] m is, 19 9 + 10 q [---------------, [0, 1, 1, 3, 3, 6, 6, 10, 10, 15, 15, 2, 2, 9, 9, 17, 17, 7, 3 2 q - q - q + 1 7, 17, 17, 9, 9, 2, 2, 15, 15, 10, 10, 6, 6, 3, 3, 1, 1], [9, 9, 18, 8, 17, 16, 6, 14, 4, 2, 11, 18, 8, 5, 14, 1, 10, 6, 15, 1, 0, 14, 3, 7, 5, 18, 6, 9, 6, 18, 5, 7, 3, 14, 0]] m is, 20 10 [------------, [0, 1, 3, 7, 13, 2, 14, 10, 10, 15, 5, 1, 3, 12, 8, 12, 4, 5, 15, 2 q - 2 q + 1 15, 5, 4, 12, 8, 12, 3, 1, 5, 15, 10, 10, 14, 2, 13, 7, 3, 1], [10, 10, 0, 10, 0, 0, 10, 0, 10, 10, 0, 10, 0, 0, 10, 0, 10, 10, 0, 10, 0, 10, 0, 10, 0, 10, 0, 10, 0, 10, 0, 10, 0, 10, 0, 10, 0]] This took, 0.368, seconds.