A Golden Penney Game

Penney's game (also known as Penney ante) is a counter-intuitive coin flip game that has attracted much attention due to Gardner's Scientific American column. We concern ourselves with just one case of Penney's game: player I choosing HHH vs. player II choosing HTH. If a trick golden penny is minted to have the probability of heads equal to 1/phi, where phi is the golden ratio, then neither player has an advantage in this game. We discover that counting the number of winning player I sequences in this game that have exactly n number of tails and k number of heads appearing before the final HHH is equivalent to counting the n-tilings of a board using exactly k fences. We derive combinatorial identities related to this counting formula, all of which are fascinating and many of which appear to be new. Some of the sequences that we encounter along the way are Pascal's triangle and a related Pascal-like triangle, the Fibonacci sequence, the Jacobsthal sequence, the golden rectangle numbers, the squared Fibonacci numbers, and more.