Journal of Integer Sequences, Vol. 24 (2021), Article 21.6.2 |

University of Bridgeport

Bridgeport, CT 06604

USA

Mark Elmer

SUNY Oswego

Oswego, NY 13126

USA

Ryan McCulloch

Elmira College

Elmira, NY 14901

USA

**Abstract:**

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/ϕ, where ϕ 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.

(Concerned with sequences A000045 A000079 A001045 A001654 A006498 A007318 A007598 A059259.)

Received December 18 2020; revised version received May 15 2021; May 16 2021.
Published in *Journal of Integer Sequences*,
May 17 2021.

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