Number of Multidimensional Spheres that can touch One Other Sphere

Theorem
Let $S_n$ be a spheres of $n$ dimensions with a given radius $r$.

Let $T \left({n}\right)$ denote the number of instances of $S_n$ that can touch one other such instance of $S_n$.

The sequence of $T \left({n}\right)$ begins as follows:


 * {| border="1"

! align="right" style = "padding: 2px 10px" | $n$ ! align="right" style = "padding: 2px 10px" | $T \left({n}\right)$
 * align="right" style = "padding: 2px 10px" | $0$
 * align="right" style = "padding: 2px 10px" | $0$
 * align="right" style = "padding: 2px 10px" | $1$
 * align="right" style = "padding: 2px 10px" | $2$
 * align="right" style = "padding: 2px 10px" | $2$
 * align="right" style = "padding: 2px 10px" | $6$
 * align="right" style = "padding: 2px 10px" | $3$
 * align="right" style = "padding: 2px 10px" | $12$
 * align="right" style = "padding: 2px 10px" | $4$
 * align="right" style = "padding: 2px 10px" | $24$
 * align="right" style = "padding: 2px 10px" | $5$
 * align="right" style = "padding: 2px 10px" | $40$
 * align="right" style = "padding: 2px 10px" | $6$
 * align="right" style = "padding: 2px 10px" | $72$
 * align="right" style = "padding: 2px 10px" | $7$
 * align="right" style = "padding: 2px 10px" | $126$
 * align="right" style = "padding: 2px 10px" | $8$
 * align="right" style = "padding: 2px 10px" | $240$
 * align="right" style = "padding: 2px 10px" | $9$
 * align="right" style = "padding: 2px 10px" | $272$
 * }
 * align="right" style = "padding: 2px 10px" | $7$
 * align="right" style = "padding: 2px 10px" | $126$
 * align="right" style = "padding: 2px 10px" | $8$
 * align="right" style = "padding: 2px 10px" | $240$
 * align="right" style = "padding: 2px 10px" | $9$
 * align="right" style = "padding: 2px 10px" | $272$
 * }
 * align="right" style = "padding: 2px 10px" | $9$
 * align="right" style = "padding: 2px 10px" | $272$
 * }