Question: I have been trying to figure out a formula for the sequence:

0, 0, 1, 1, 2, 2, 3, 3, 4, 4

It is not geometric and it is not arithmetic, I tried to apply these formulas, but of them are failing leading me to believe it is not arithmetic nor geometric, can this be possibe? Is there another type of sequence?

Arithmetic:

[![enter image description here](https://i.stack.imgur.com/RR8y0.png)](https://i.stack.imgur.com/RR8y0.png)

Geometric:

[![enter image description here](https://i.stack.imgur.com/9Slkh.png)](https://i.stack.imgur.com/9Slkh.png)

Answer: 
Consider the following counterexample: let $V = \mathbb{R}^3$ and $W = span\{(1, 0,0), (0, 1, 0) \}$. Take $w\_1 = \frac{1}{\sqrt{2}} (1, 1, 0)$, $w\_2 = (1, 0, 0)$ and $v = (1,1,1)$. Then 
\begin{equation}
v - P\_W(v) = (1, 1, 1) - (1, 1, 0) - (1, 0, 0) = (-1, 0, 1)
\end{equation}
and we see that it is not perpendicular to $W$. The right definition for $P\_W$ would be the one that first applies the Gram-Schmidt procedure to the basis for W and then plugs that new, orthonormal basis into your formula.
This formula for the orthogonal projection is valid iff the basis is orthogonal. Clearly it is valid if the basis is orthogonal. Assume now the formula is valid, then we get
$$ w\_i= P\_W(w\_i) = \sum\_{k=1}^n \frac{\langle w\_i, w\_k \rangle}{\langle w\_k, w\_k \rangle} w\_k = w\_i + \sum\_{k=1, k\neq i}^n \frac{\langle w\_i, w\_k \rangle}{\langle w\_k, w\_k \rangle} w\_k $$
But the $w\_j$'s are independent, thus we have for any $1\leq k \leq n$ with $k\neq i$
$$ \frac{\langle w\_i, w\_k \rangle}{\langle w\_k, w\_k \rangle}=0 $$
and hence
$$ \langle w\_i, w\_k \rangle =0 $$
Therefore, our basis is orthogonal.