[From V. Raman, Dec 12 2012] (Start)
The Old Maid match is played with four players. For every game that is played, one of the four players is trolled at the end. The match ends when a player has been trolled three times, after which he loses the match.
So, to calculate the probability, we represent the four players by the digits 0, 1, 2, 3 in base 4 and then list out all the 18-bit numbers in base 4.
Then a(n) = Number of numbers for which from the left, some digit has occurred three times at the n-th position, i.e. 32 * A220017(n).
For four player match, the maximum number of games needed for a player to be trolled three times is 9. So, we consider with 9-digit base 4 numbers.
Total value of a(i), for i = 3..9 is equal to 4^9 = 262144.
Total value of A220017(i), for i = 3..9 is equal to 4^9/32 = 262144/32 = 8192. (relative chance)
gcd of a(i), for i = 3..9 is equal to 2^5 = 32.
The sequence A220017 gives the relative probability for the match to last exactly for n games. (End)