c(n): the number of connectors at level n;
C(n): the sum of all number of connectors up to level n,
i.e., C(n) = c(n) + c(n-1) + … + c(1) + c(0)
S(n): the number of sticks to construct n-level Tetrahedron.

Let’s observe the easy ones first:
n=0: c(0)=1, C(0)=1, S(0)=0;
n=1: c(1)=3, C(1)=c(1)+c(0)=4, S(1)=6
n=2: c(2)=6, C(2)=c(2)+c(1)+c(0)=10, S(2)=24

the pattern of c(n) can be formulated as:
c(n) = c(n-1) + (n+1)

the patter of C(n) can be formulated as:
C(n) = C(n-1) + c(n)

the pattern of S(n) can be formulated as:
S(n) = C(n-1) * 6

According to above three formula, here are the answers for level 1 to level 10:

So the answer for a 5 level Tetrahedron will require 56 connectors and 210 sticks, and 10 level one will require 286 connectors and 1320 sticks if you ever build such a gigantic Tetrahedron.