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We define $$S(n)=\sum_{a=2+(n\pmod 2)}^{n-2} \sharp(\{j,1\leq j<n \pmod{a},(a,j)=1\})\ .$$ (Searching the OEIS yielded no results.) For $n>2$ we have the following experimental observations (...

Here is a new conjecture of mine from the appendix of an unpublished manuscript currently under review. Let $b \in \mathbb{Z}^+$ and assume that $n$ is an integer greater than $1$ and not a multiple ...

The Domb numbers are given by $$D_n=\sum_{k=0}^n\binom{n}{k}^2\binom{2k}k\binom{2(n-k)}{n-k}\ \ \ (n=0,1,2,\ldots).$$ Such numbers have combinatorial interpretation, see, e.g., http://oeis.org/A002895....