Letting $z=e^{i\theta}$, the Fourier expansion ${1\over 2\pi} \sum_{n\in \mathbb Z} 1\cdot e^{in\theta}$ is the Fourier expansion of a "Dirac comb", meaning a sum of Dirac delta distributions at integer multiples of $2\pi$. This Fourier series converges perfectly well in a Sobolev-like space with Hilbert-space norm
$$
\Big|\sum_n c_n\,e^{in\theta}\Big|_s^2 \;=\; \sum_n |c_n|^2\cdot (1+n^2)^s
$$
for $s<-1/2$. Indeed, for sufficiently nice functions, the fact that their Fourier series converge pointwise to them can be interpreted as verifying that this distribution behaves as claimed.
In particular, the fact that this distribution really is locally $0$ away from the Dirac-delta spikes is a sort of certification that it "is zero" there. (Distributions are localizable.)
That is, the Fourier series of (periodic) distributions certainly do not converge pointwise, just as Fourier transforms of tempered distributions often do not. But such things do certainly converge in a different topology, and are enormously useful.
Essentially, on average, how many digits of pi would you have to have to go through to find a specific sequence of a given length?
If the statistical behavior of the digits of $\pi$ is modeled by an i.i.d. sequence of random variables uniform on $\{0,1,2,3,4,5,6,7,8,9\}$, the following theorem is relevant:
If $(X_1, X_2, X_3, \dots)$ is a sequence of random variables that are i.i.d. uniform on a set $S$ of $m$ elements, then the expected first-occurrence time of a contiguous subsequence $\tau\in S^n$ is given recursively by
$$ET_\tau = m^n + ET_\sigma
$$
where $\sigma$ is the longest proper suffix of $\tau$ that is also a prefix of $\tau$. Here $T_\tau$ is defined as the least $k$ such that $\tau$ is a contiguous subsequence of $(X_1, \dots , X_k)$, and by convention $T_{\text{empty sequence}}=0$.
(A proof of this is in the textbook by Sheldon M. Ross, "Introduction to Probability Models", 10th Edition, Example 4.26, pp 225-228. Ross proves a much more general result for Markov chains.)
E.g., with $m = 10$,
$$\begin{align}
ET_{01201201} & = 10^8 + ET_{01201}\\
& = 10^8 + 10^5 + ET_{01}\\
& = 10^8 + 10^5 + 10^2 + ET_{\text{empty sequence}}\\
& = 10^8 + 10^5 + 10^2
\end{align}
$$
whereas the first-occurrence time of this sequence in the digits of $\pi$ is $91,997,854 \approx 0.9 ET_{01201201}$.
Some observations:
A bounding case is when $\tau$ is composed of $n$ copies of a single symbol, e.g. $0^n = 000...0$, so
$$\begin{align}
ET_{0^n} & = m^n + ET_{0^{n-1}}\\
& = m^n + m^{n-1} + ET_{0^{n-2}}\\
& ...\\
& = m^n + m^{n-1} + ... + 1\\
& = \frac{m^{n+1} - m}{m-1}
\end{align}
$$
Hence, for all $\tau$,
$$m^n \le ET_\tau \le \frac{m^{n+1} - m}{m-1} \lt m^{n+1}
$$
so the growth is exponential in subsequence-length $n$.
Interestingly, $ET_\tau$ is always an integer.
Best Answer
Test if $a(n)$ is a polynomial in $n$ [$a(n)$ denotes the $n^{th}$ term]. In other words, are the differences of some order constant?
Test if the differences of some order are periodic. (Suppose the $k^{th}$ order differences are $d(1), ...,d(n)$. They are said to be periodic if there is a number $p$, the period, with $1 \le p \le n-2$, such that $d(i) = d(j)$ whenever $i = j \pmod{p}$.
Test if any row of the difference table of some depth is essentially constant. This detects such sequences as $4^n - n^4$. (Let the usual difference table be $a(0), a(1), a(2), \cdots $, $b(0), b(1), \cdots$, $c(0), c(1), \cdots , /cdots $. This is the difference table of depth $1$. The table of depth $2$ has as top row $a(0), b(0), c(0), \cdots $ and so on.
For a $2$-valued sequence, compute the six characteristic sequences associated with the sequence and look them up in the OEIS.(Suppose the sequence takes only the values $X$ and $Y$. The six characteristic sequences, all equivalent to the original, are: replace $X,Y$ by $1,2$; by $2,1$; the positions of the $X's$, of the $Y's$; the run lengths; and the derivative, i.e. the positions where the sequence changes.
Form the generating functions (g.f.) for the sequence for each of the following $6$ types: ogf ordinary generating function, egf exponential generating function, revogf reversion of ordinary generating function, revegf reversion of exponential generating function, lgdogf logarithmic derivative of ordinary generating function, lgdegf logarithmic derivative of exponential generating function, and attempt to represent them as rational functions, hypergeometric series, or the solution to a linear differential equation with polynomial coefficients.
Look for a linear recurrence with polynomial coefficients for the coefficients of the above $6$ types of g.f.'s.
Look for a polynomial equation in $y$ and $x$ for the g.f. y(x) of each of the above $6$ types.
Apply the transformations listed below to the sequence and look up the result in the OEIS. Stop when $50$ matches have been found. (Note: there is a very long list of transformations in the help page).
Test if the sequence is a Beatty sequence. (A Beatty sequence is one in which the $n^{th}$ term is $[nz]$, where $z$ is irrational. The complementary sequence is $[ny]$, where $\frac{1}{x} + \frac{1}{y} = 1$. Refs: N. J. A. Sloane, Handbook of Integer Sequences, $1973$, p. $29$; R. Honsberger, Ingenuity in Mathematics, $1970$, p. $93$. If this is a Beatty sequence the value given for $z$ will produce the given terms, but this value of $z$ is very far from being unique.)
The list continues with transformations and other types of possible recurrences. It provides as well a long list of mathematical packages very useful for pattern recognition.