If the sequence $a_1,a_2,a_3,…$ is periodic, show that $a_1,a_2,a_9,a_{10}, …=a_{n^2-n+1+(-1)^{n-1}(n-1)}\; , \; n\in \mathbb{N}$ is also periodic

Question

If the sequence $t_i = a_1,a_2,a_3,…$ with $a_i\in \{2,3,5,7\}$ is periodic, show that $s_n = a_1,a_2,a_9,a_{10},a_{25},a_{26},… = a_{n^2-n+1+(-1)^{n-1}(n-1)}\; , \; n\in \mathbb{N}$ is also periodic.

For example, if $t_i = a_1,a_2,a_3,a_4,a_5,a_6,… = 2,3,2,3,2,3,…$

$s_n = a_1,a_2,a_9,a_{10},a_{25},a_{26},… = 2,3,2,3,2,3,…$

So in this special case, they are even equal.

But what about all the other periodicities of $t_i$ one can construct with $2,3,5,7$? Especially if the least period is $>2$