This can't be a coincidence as well

Consider the decimal expansion of 1/9989989991 / 998998999:

0.000000001001002  0040070130240. \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 {an}\lbrace a_{n} \rbrace - satisfy an=an1+an2+an3a_{n} = a_{n-1} + a_{n-2} + a_{n-3}.

If a1=a2=1a_{1} = a_{2} = 1 and a3=2a_{3} = 2, find the least nn for which the recurrence is no longer satisfied.

Inspired by this problem.

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