The trauma I went through on Problem 64 made this one a breeze. I also used BigFractions class from this website.

public String algorithm ()
{
int [] a = new int [100];
a[0] = 2;
int k = 1;
for (int i = 1; i <= 99; i=i+3)
{
if (i <= 99)
a[i] = 1;
if (i+1 <= 99)
a[i+1] = 2*k++;
if (i+2 <= 99)
a[i+2] = 1;
}
BigFraction F = new BigFraction(a[99],1);
for (int i = 99; i >= 1; i--)
{
F = F.inverse();
F = F.add(a[i-1]);
}
int sum = 0;
String num = F.numerator().toString();
for (int n = 0; n <= num.length()-1; n++)
sum += Integer.parseInt(num.substring(n, n+1));
return String.format("\nAnswer = %s\n", String.valueOf(sum));
}

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