Linear Recurrence With Little Information

Probability Level 4

An infinite sequence of real numbers a1,a2,a_1,a_2,\ldots satisfies the recurrence an+3=an+22an+1+ana_{n+3}=a_{n+2}-2a_{n+1}+a_n for every positive integer nn.

Given that a1=a3=1a_1=a_3=1 and a98=a99a_{98}=a_{99}, compute a1+a2++a100a_1+a_2+\cdots+a_{100}.


Source: HMMT.
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