<s>[INST] Is the structure of sequence A094615 obvious? Is the triangular structure described in A094615 proven?
That is, is it proved that the following computation lasts forever?
\begin{eqnarray*}
3 &=& 2\cdot 1 + 1 \\
5 &=& 3\cdot 1 + 2 \\
\ \\
7 &=& 2\cdot 3 + 1 \\
11 &=& 2\cdot 5 + 1 = 3\cdot 3 + 2\\
17 &=& 3\cdot 5 + 2 \\
\ \\
15 &=& 2\cdot 7 + 1 \\
23 &=& 2\cdot 11 + 1 = 3\cdot 7 + 2\\
35 &=& 2\cdot 17 + 1 = 3\cdot 11 + 2\\
53 &=& 3\cdot 17 + 2\\
\vdots
\end{eqnarray*}
Thank you.
 [/INST] That linked page has an explicit representation of the entries given by Lamine Ngom, namely
$$T(n,k) = 2^{n+1-k}3^{k-1} -1 $$
Should be not too hard to verify by induction that the triangular pattern continues.
</s>