Palindromic Infinite Sequence

An infinite sequence $\{S_1,S_2,S_3,S_4,\dots \}$ is said palindromic if:
for every $n_0$ there is $n\ge n_0$ such as $\{S_1,S_2,S_3,S_4,\dots ,S_n\}$ is palindromic.

It can be proved that for any irrational $\alpha$, the sequence $(\lfloor \alpha \times n)-(\lfloor \alpha \times (n-1))$ for $n\ge 1$ is palindromic.
$\lfloor (x)$ is the integer part of x.

For the irrational $e=2.71828\dots$, this sequence is palindromic for following indexes:

• 1 - 1
• 2 - 4
• 3 - 11
• 4 - 18
• 5 - 25
• 6 - 32
• 7 - 71

Find the ${10}^{9th}$ index. Give your answer modulo ${10}^{10}$

You are given 3352820698 the ${10}^{2th}$ index

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