This can't be a coincidence as well

Number Theory Level 5

Consider the decimal expansion of $1 / 998998999$ :

$0. \quad 000 \quad 000 \quad 001 \quad 001 \quad 002 \\ \ \ \quad 004 \quad 007 \quad 013 \quad 024 \quad \ldots$

Notice that those numbers in units of three digits - let's call them $\lbrace a_{n} \rbrace$ - satisfy $a_{n} = a_{n-1} + a_{n-2} + a_{n-3}$ .

If $a_{1} = a_{2} = 1$ and $a_{3} = 2$ , find the least $n$ for which the recurrence is no longer satisfied.

Inspired by this problem.

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