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Here are collected some results related with the operator $a$ which commutation with its adjoint $\ud{a}$ reads:
$$\tag{1}[a,\ud{a}]=1$$
(cf. Fermi algebra)
I have written a Mathematica piece of code to compute such correlators automatically. I found out in this way formulas already known in the literature, in particular from the work of Blasiak^{[1]}^{[2]}. You can download it to play with it (check or compute results useful in your daily quantum algebra). This is detailed in this blog post.
The basic rules are:
$$ \begin{align*} a\ket{n}&=\sqrt{n}\ket{n1}\,,&\bra{n}\,&a=\bra{n+1}\sqrt{n+1}\,,\\ \ud{a}\ket{n}&=\sqrt{n+1}\ket{n+1}\,,&\bra{n}\,&\ud{a}=\bra{n1}\sqrt{n}\,, \end{align*} $$
of which a general expression can be drawn:
$$a^{\dagger i}a^j a^{\dagger k}\ket{n}={(n+k)!\over(n+kj)!}\sqrt{(n+i+kj)!\over n!}\ket{n+i+kj}\,.$$
Some particular cases:
$$ \begin{align*} \kern1cm{(\mathrm{for}~i\le n+j)}\kern1cm a^i{\ud{a}}^j\ket{n}&={(n+j)!\over\sqrt{n!}\sqrt{(n+ji)!}}\ket{n+ji}\,,\\ \kern1cm{(\mathrm{for}~i\le n)}\kern1cm a^{\dagger j}a^i\ket{n}&={\sqrt{n!}\sqrt{(n+ji)!}\over(ni)!}\ket{n+ji}\,. \end{align*} $$
A general result, based on Wick theorem:
$$a^n\ud{a}^m=\sum_{k=0}^{\min(n,m)}k!\binom{m}{k}\binom{n}{k}\ud{a}^{mk}a^{nk}$$
Another one, inferred from my Mathematica notebook above, that could certainly be proved by recurrence or directly from combinatorics:
$$(a\ud{a})^n=\sum_{k=0}^n \left\{\begin{matrix} n\\k \end{matrix} \right\} \ud{a}^ka^k$$
where $\left\{\begin{matrix} n\\k \end{matrix} \right\}$ are the Stirling partition numbers (the number of ways to partition a set of $n$ objects into $k$ nonempty subsets). It is easy to obtain similar results in terms of generalized Stirling and Bell numbers for $(a^k\ud{l})^n$, which are results already provided by Blasiak.
$$a\ud{a}^n=\ud{a}^na+n\ud{a}^{n1}$$ $$a^n\ud{a}=\ud{a}a^n+na^{n1}$$
A fairly general result (forall $k, l\in\mathbb{N}$, including $0$):
$$aa^{\dagger k}a^la^\dagger=a^{\dagger(k+1)}a^{l+1}+(k+l+1)a^{\dagger k}a^l+kla^{\dagger(k1)}a^{l1}$$
I maintain a long list of Bose algebra expressions in canonic notations to assist me in my computations. You can glance at it to find particular and redundant cases of the above.
A compilation of useful results derived from Eq.~(1):
We define $\hat n\equiv\ud{a}a$.

Special cases below
