Numbers that are not the sum of 3 generalized 7-gonal numbers.
10, 16, 76, 307
1
It appears that all other positive numbers are the sum of 3 generalized 7-gonal numbers. The sum of 3 generalized pentagonal numbers or 3 generalized hexagonal numbers can represent any positive number; there are no exceptions.
T. D. Noe, Plot of 4 terms
Eric W. Weisstein, MathWorld: Polygonal Number
(Mma) f = 7; nn = 50; t = Table[k ((f - 2) k - (f - 4))/2, {k, -nn, nn}]; t2 = Select[Union[Flatten[Table[a + b + c, {a, t}, {b, t}, {c, t}]]], # <= t[[-1]] &]; Complement[Range[t[[-1]]], t2]
nonn,fini
T. D. Noe, Dec 28 2015