a_{p-1}=2^{p-1}+3^{p-1}+6^{p-1}-1=\\
=2\cdot 2^{p-2}+3\cdot 3^{p-2}+6\cdot 6^{p-2}-1=\\
=a_{p-2}+2^{p-2}+2\cdot 3^{p-2}+(2+3)\cdot 6^{p-2}=\\
=a_{p-2}+2^{p-2}+2\cdot 3^{p-2}+2\cdot 6^{p-2}+3\cdot 6^{p-2}=\\
={a_{p-2}+2^{p-2}+2\cdot 3^{p-2}+2\cdot 2^{p-2}\cdot 3^{p-2}+3\cdot 2^{p-2}\cdot 3^{p-2}}\equiv \\
\equiv{a_{p-2}+2^{p-2}+2\cdot 3^{p-2}+3^{p-2}+2^{p-2}}\equiv \\
\equiv{a_{p-2}+2\cdot 2^{p-2}+3\cdot 3^{p-2}}\equiv \\
\equiv{a_{p-2}+1+1}\equiv \\
\equiv{a_{p-2}+2}\pmod p