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Some summtion formulas like $\sum_{i=1}^{n}i = \frac{n(n+1)}{2}$ and,

$\sum_{i=1}^n\frac{i}{2^i} = 2- \frac{n+2}{2^n}$ hold and can be proven by induction.

However, I'm curious to know if there are any formulas that hold true only for the first few terms ($n \in [1,50]$), but fail to hold for larger values of $n$.

Thank you in advance

MR.-c
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  • "Formulas" is too wide a concept. Nearly everything is a formula in some language. For a sufficiently wide meaning you can put on the left many examples. Say, the number of coefficients of the cyclotomic polynomial of degree $n$ of absolute value larger than $1$, and on the right put $0$. This formula holds for $n$ up to $104$. – NDB Jul 27 '23 at 14:29
  • There are interesting restrictions that can be considered that present the opposite phenomenon. For example, if the sides are constant recursive sequences, then there is a finite number of terms to consider, that if they match, then the equation is true for all $n$. – NDB Jul 27 '23 at 14:32
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    There is the famous example of Borwein's integrals (https://math.stackexchange.com/q/3025772/399263), they can be modified to fail at larger $n$ than original $n=15$. 3b1b have a good video on these. – zwim Jul 27 '23 at 19:06
  • You may find many examples for $n \le 10$ if you take a simple function, compute first several elements and search at the Online Encyclopedia of Integer Sequences. For example I've found right now that the number of distinct sums $i^3 + j^3$ for $1 \le i \le j \le n$ equals to $\frac{n(n + 1)}{2}$ for $n \le 11$. – Smylic Aug 02 '23 at 15:02

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