The Generalized Capsules for the coefficient of q^(m*n+1) mod m for the unique formal power series satisying thefunctional equation m q f(q ) f(q) = 1 + -------- 2 (1 - q) for m from 3 to 20, except those that are 2 mod 4 are as follows m is, 3 1 [- ------, [0, 1, 0, 2], [1, 1, 1, 2]] 2 q - 1 m is, 4 FAIL m is, 5 1 [- ------, [0, 1, 0, 2, 0, 3, 0, 4], [1, 1, 1, 2, 1, 3, 2, 4]] 2 q - 1 m is, 6 FAIL m is, 7 1 [- ------, [0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6], 2 q - 1 [1, 1, 1, 2, 1, 3, 1, 4, 2, 5, 3, 6]] m is, 8 FAIL m is, 9 1 [- ------, [0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, 0, 7, 0, 8], 2 q - 1 [1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 2, 6, 3, 7, 4, 8]] m is, 10 FAIL m is, 11 1 [- ------, [0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, 0, 7, 0, 8, 0, 9, 0, 10], 2 q - 1 [1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 2, 7, 3, 8, 4, 9, 5, 10]] m is, 12 FAIL m is, 13 1 [- ------, 2 q - 1 [0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, 0, 7, 0, 8, 0, 9, 0, 10, 0, 11, 0, 12] , [1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 2, 8, 3, 9, 4, 10, 5, 11, 6, 12] ] m is, 14 FAIL m is, 15 1 [- ------, [0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, 0, 7, 0, 8, 0, 9, 0, 10, 0, 11, 2 q - 1 0, 12, 0, 13, 0, 14], [1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 2, 9, 3, 10, 4, 11, 5, 12, 6, 13, 7, 14]] m is, 16 FAIL m is, 17 1 [- ------, [0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, 0, 7, 0, 8, 0, 9, 0, 10, 0, 11, 2 q - 1 0, 12, 0, 13, 0, 14, 0, 15, 0, 16], [1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 10, 3, 11, 4, 12, 5, 13, 6, 14, 7, 15, 8, 16]] m is, 18 FAIL m is, 19 1 [- ------, [0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, 0, 7, 0, 8, 0, 9, 0, 10, 0, 11, 2 q - 1 0, 12, 0, 13, 0, 14, 0, 15, 0, 16, 0, 17, 0, 18], [1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 1, 10, 2, 11, 3, 12, 4, 13, 5, 14, 6, 15, 7, 16, 8, 17, 9, 18]] m is, 20 FAIL This took, 0.154, seconds.