Is there a notation for $(0,...,0,k,0, ...,0)$? Or in general: $(a,...,a,k,a,...,a)$? Suppose that we want to define the canonical basis of $\mathbb R^n$. I find it quite annoying and not too precise to say: "$e_k = (0,...,0,1,0,...,0)$ where $1$ is in the $k$th position", because this assumes that $k \ge 3$ and $\le n-2$.
Thanks.
I'd write $(0,\dots,0,k,0,\dots,0)$ as $ke_i$, with $e_i$ the appropriate unit vector.
Since you say that the "naive" definition $e_i = (0,\dots,1,\dots,0)$ is not appealing to you, consider $\Bbb R^n$ as the functions $\{1,\dots,n\}\to\Bbb R$ and define:
$$e_i(j) = \delta_{ij}$$
where $\delta_{ij}$ is the Kronecker delta: $1$ if $i=j$ and $0$ otherwise.
Another notation that might be handy is $\mathbf 1$, defined by $\mathbf 1(i) = 1$. Using this, we can write:
$$(a,\dots,k,\dots,a) = a\mathbf 1 + (k-a)e_i$$