Counting The Sixes

A fair, \(6\)-sided die is rolled \(20\) times, and the sequence of the rolls is recorded.

\(C\) is the number of times in the 20-number sequence that a subsequence (of any length from one to six) of rolls adds up to \(6.\) These subsequences don't have to be separate and can overlap each other. For example, the sequence of \(20\) rolls \[ 12334222111366141523 \] contains the ten subsequences \(123, 33, 42, 222, 2211, 1113, 6, 6, 141, 15\) which all add up to \(6,\) so \(C=10\) in this case.

The expected value of \(C\) is equal to \(\frac{a}{b}\) for coprime positive integers \(a\) and \(b.\)

What is \(a+b?\)

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