Reminder Problem (Self-Made)

[Phlembac Adib Hasan]

($10000^{th}$ post)A sequence is defined in this way:
\[a_0=3,a_1=7,a_n=(a_{n-1})^{n^n}+(a_{n-2})^{n^n}\; \; \; \text {for }n\ge 2\]Find the reminder if $a_{200000}$ is divided by $4$.

(Edited for correcting the spellings,not to change the problem.Actually I am not so good in English and also in typing.)

[sourav das]

($10000^{th}$ post)A sequence is defines in this way:
\[a_0=3,a_1=7,a_n=(a_{n-1})^{n^n}+(a_{n-2})^{n^n}\; \; \; \text {for }n\ge 2\]Find the reminder if $a_{200000}$ is divided by $4$.

Please clear the statement. What is divisible by 4 and remainder of what?

[Phlembac Adib Hasan]

($10000^{th}$ post)A sequence is defines in this way:
\[a_0=3,a_1=7,a_n=(a_{n-1})^{n^n}+(a_{n-2})^{n^n}\; \; \; \text {for }n\ge 2\]Find the reminder if $a_{200000}$ is divided by $4$.

Please clear the statement. What is divisible by 4 and remainder of what?

$a_{200000}\equiv ?(mod\; 4)$
Actually I chose $200000$ at random.I can choose anything else.Like $a_{124563}\equiv ?(mod\; 4)$.

[*Mahi*]

$a_6 \equiv 1 \pmod {4}$, $a_7 \equiv 1 \pmod 4$ and $a_8 \equiv 2 \pmod 4$. This with the fact $1^x \equiv 1 \pmod 4$ and $2^{k} \equiv 0 \pmod 4$ for $k>1$ gives the solution.