Leet Sum

Algebra Level 5

The sequence {ak}k=1112\{a_k\}_{k=1}^{112} satisfies a1=1a_1 = 1 and an=1337+nan1a_n = \dfrac{1337 + n}{a_{n-1}}, for all positive integers nn. Let

S=a10a13+a11a14+a12a15++a109a112.S = \left\lfloor a_{10}a_{13} + a_{11}a_{14} + a_{12}a_{15} + \cdots + a_{109}a_{112}\right\rfloor.

Find the remainder when SS is divided by 10001000.

This problem is posed by Akshaj.

Details and assumptions

The function x:RZ\lfloor x \rfloor: \mathbb{R} \rightarrow \mathbb{Z} refers to the greatest integer smaller than or equal to xx. For example 2.3=2\lfloor 2.3 \rfloor = 2 and 5=5\lfloor -5 \rfloor = -5.

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