So Many Chairs, No???

Level pending

There are \(14\) chairs arranged in a line. Let \(T_k\) denote the number of ways to choose \(k\) chairs from them such that no two of the chosen chairs are consecutive. Evaluate: \[\sum_{k=0}^{14} T_k\] Try to generalize for \(n\) chairs.

Image Credit: Origin Furniture

EDIT: I'm very sorry as there was a problem in the problem (!!!), fixed!!! :D

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