PROJECT EULER #65

Link to Project Euler problem 65

The square root of 2 can be written as an infinite continued fraction.

2 = 1 +	1
	2 +	1
		2 +	1
			2 +	1
				2 + ...

The infinite continued fraction can be written, 2 = [1;(2)], (2) indicates that 2 repeats ad infinitum. In a similar way, 23 = [4;(1,3,1,8)].

It turns out that the sequence of partial values of continued fractions for square roots provide the best rational approximations. Let us consider the convergents for 2.

1 +	1	= 3/2
	2

1 +	1		= 7/5
	2 +	1
		2

1 +	1			= 17/12
	2 +	1
		2 +	1
			2

1 +	1				= 41/29
	2 +	1
		2 +	1
			2 +	1
				2

Hence the sequence of the first ten convergents for 2 are:

1, 3/2, 7/5, 17/12, 41/29, 99/70, 239/169, 577/408, 1393/985, 3363/2378, ...

What is most surprising is that the important mathematical constant,
e = [2; 1,2,1, 1,4,1, 1,6,1 , ... , 1,2k,1, ...].

The first ten terms in the sequence of convergents for e are:

2, 3, 8/3, 11/4, 19/7, 87/32, 106/39, 193/71, 1264/465, 1457/536, ...

The sum of digits in the numerator of the 10^th convergent is 1+4+5+7=17.

Find the sum of digits in the numerator of the 100^th convergent of the continued fraction for e.

using System;

namespace project_euler
{
  class Program
  {
      static void Main()
      {
          //Problem 65
          DateTime start = DateTime.Now;
          //e = [2;1,2,1,1,4,1,1,6,1,....,1,2k,1,...]
          //numerator_n = a_n * numerator_n-1 + numerator_n-2
          //denominator_n = a_n * denominator_n-1 + denominator_n-2
          BigInt numerator_2 = 2, numerator_1 = 3, numerator=0;
          //BigInt denominator_1 = 1,denominator_2 = 1, denominator;
          int    sum = 0;
          for (int i = 2; i < 100; i++)
          {
              int a = (i + 1) % 3 == 0 ? 2 * ((i + 1) / 3) : 1;
              numerator = numerator_1 * a + numerator_2;
              numerator_2 = numerator_1;
              numerator_1 = numerator;
              //denominator = a * denominator_1 + denominator_2;
              //denominator_2 = denominator_1;
              //denominator_1 = denominator;
              //Console.WriteLine(numerator);
              //Console.WriteLine("-----------------------------------------");
              //Console.WriteLine(denominator);
          }
          char[] c = numerator.ToString().ToCharArray();
          foreach (char c1 in c)
              sum += int.Parse(c1.ToString());
          Console.WriteLine(sum);
          TimeSpan time = DateTime.Now - start;
          Console.WriteLine("This took {0}", time);
          Console.ReadKey();
      }
  }
}