Yes, basically any F(n), polynomial or not, such that F(1) = 3/17, ..., F(5) = 4e, as long as it is valid for all natural numbers n, is a potential source of such answers. The first 5 terms will be the same as in our sequence, but the 6th one will differ in general. Consider this slightly altered clandarkfire's formula:
F(n,p,r,s,t) = (3/17)sin[(n-2)(n-3)(n-4)(n-5)]/sin(24) + (22.5)[(n-1)(n-3)(n-4)(n-5)/-6]^p + (-15)[(n-1)(n-2)(n-4)(n-5)/4]^r + (1.25pi)[(n-1)(n-2)(n-3)(n-5)/-6]^s + (4e)[(n-1)(n-2)(n-3)(n-4)/24]^t
Setting n=6 and p=r=s=t=1 gives clandarkfire's answer, although from different (and non-polynomial) formula. For other (allowed) p, r, s, t values it will give different formula's, and, in general, different answers, but the first 5 terms will remain the same in each case.
Perhaps an easier way to think of it is as follows. If the formula for 5 terms can be constructed, then it can be done for 6 (and 7, 8, ...). So we can pick some arbitrary real number R, and build the new formula using the same sneaky Lagrange polynomials:
G(n) = (3/17)(n-2)(n-3)(n-4)(n-5)(n-6)/-120 + (22.5)(n-1)(n-3)(n-4)(n-5)(n-6)/24 + (-15)(n-1)(n-2)(n-4)(n-5)(n-6)/-12 + (1.25pi)(n-1)(n-2)(n-3)(n-5)(n-6)/12 + (4e)(n-1)(n-2)(n-3)(n-4)(n-6)/-24 + R(n-1)(n-2)(n-3)(n-4)(n-5)/120
Then G(1) = 3/17, ..., G(5) = 4e, G(6) = R, ...
But R is arbitrary - so, in fact, any real number can be the answer to the OP problem.
This can be applied to any "What is the next term in this or that sequence" type problem. For instance, what is the next term in the sequence F(1) = 1, F(2) = 2, F(3) = 3 ? One might think that F(n) = n, and the answer is 4. But what about
F(n) = (n-2)(n-3)(n-4)/-6 + 2*(n-1)(n-3)(n-4)/2 + 3*(n-1)(n-2)(n-4)/-2 + 7.34*(n-1)(n-2)(n-3)/6 ?
Now F(1) = 1, F(2) = 2, F(3) = 3, but F(4) = 7.34. F(4) can be any number, using the same argument as above.
This is a bit confusing, it has to be said. Perhaps a way I would try to explain it is this. When we deal with non-random number sequences in some mathematical context, we expect them to be unique. To assure us of that we are usually given the index, with which the sequence begins, and a rule, from which the sequence can be determined uniquely - a formula for a general term, a recurrence relation, a generating function for that sequence, whatever. And if there is no such rule specified, then the sequence must be random? In which case, the next term is just a random number.