Frequently in mathematics the best way to determine the value of a sequence at a particular index is to compute its value at every index, even though the latter seems on the surface like a harder problem.
Here is one of my favorite examples of this phenomenon. Suppose you want to know how many vectors of a particular norm there are in some lattice $L$. On the surface, this seems like a hard problem - it involves figuring out how many times some quadratic form takes some value. One can solve this problem by solving the harder problem of determining the answer for every possible norm by writing down the theta function $$\Theta_L(\tau) = \sum_{v \in L} e^{-\pi \tau \left< v, v \right>}.$$$$\Theta_L(\tau) = \sum_{v \in L} e^{\pi i \tau \left< v, v \right>}.$$
If $L$ satisfies certain technical properties, $\Theta_L$ is a modular form with respect to some congruence subgroup, and modular forms live in finite-dimensional vector spaces; moreover, a lot is known about how to write down modular forms. For example, the theta function of the $E_8$ lattice is a modular form of weight $4$ and level $1$. The space of such forms is one-dimensional - in fact, it's spanned by an Eisenstein series - and it then follows that $$\Theta_{E_8}(\tau) = 1 + 240 \sum_{n \ge 1} \sigma_3(n) q^n$$
where $q = e^{2\pi i \tau}$. Similar considerations lead to the well-known formulas for the number of ways to represent an integer as the sum of two or four squares.