{"app":{"locale":"en-GB"},"problem":{"content":"\rAn infinite sequence of real numbers a(n) is defined for all integers n as follows:$$a(n) = \\begin{cases}1 & n \\lt 0\\\\\\sum \\limits_{i = 1}^{\\infty}{\\dfrac{a(n - i)}{i!}} & n \\ge 0\\end{cases}$$

For example,

a(0) =

+ ... = e − 1

a(1) =

e − 1

+ ... = 2e − 3

a(2) =

2e − 3

e − 1

+ ... =

e − 6

with e = 2.7182818... being Euler's constant.

It can be shown that a(n) is of the form

A(n) e + B(n)

n!

for integers A(n) and B(n).

For example a(10) =

328161643 e − 652694486

10!

Find A(10⁹) + B(10⁹) and give your answer mod 77 777 777.

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