yaworsw/euler-manager

---
:id: 330
:name: Euler's Number
:url: https://projecteuler.net/problem=330
:content: "\r An infinite sequence of real numbers <var>a</var>(<var>n</var>) is defined
  for all integers <var>n</var> as follows:\n ![p330_formula.gif]({{ images_dir }}/p330_formula.gif)\n\nFor
  example,\n\n| <var>a</var>(0) = | \n\n| 1 |\n| 1! |\n\n | + | \n\n| 1 |\n| 2! |\n\n
  | + | \n\n| 1 |\n| 3! |\n\n | + ... = e − 1 |\n\n| <var>a</var>(1) = | \n\n| e −
  1 |\n| 1! |\n\n | + | \n\n| 1 |\n| 2! |\n\n | + | \n\n| 1 |\n| 3! |\n\n | + ...
  = 2e − 3 |\n\n| <var>a</var>(2) = | \n\n| 2e − 3 |\n| 1! |\n\n | + | \n\n| e − 1
  |\n| 2! |\n\n | + | \n\n| 1 |\n| 3! |\n\n | + ... = | \n\n| 7 |\n| 2 |\n\n | e −
  6 |\n\nwith e = 2.7182818... being Euler's constant.\n\n| It can be shown that <var>a</var>(<var>n</var>)
  is of the form | \n\n| A(<var>n</var>) e + B(<var>n</var>) |\n| <var>n</var>! |\n\n
  | for integers A(<var>n</var>) and B(<var>n</var>). |\n\n| For example <var>a</var>(10)
  = | \n\n| 328161643 e − 652694486 |\n| 10! |\n\n | . |\n\nFind A(10<sup>9</sup>)
  + B(10<sup>9</sup>) and give your answer mod 77 777 777.\n\n"