A coin flip game and generalizations of Fibonacci numbers

TL;DR

This paper analyzes a coin flip game where the expected number of flips until a specific pattern appears is calculated, revealing new summation identities related to generalized Fibonacci numbers.

Contribution

It introduces a detailed analysis of a coin flip game with specific pattern constraints and derives new summation identities involving generalized Fibonacci numbers.

Findings

Expected number of flips for patterns with up to four maximal runs calculated

Derived new summation identities involving generalized Fibonacci numbers

Extended Fibonacci concepts to analyze pattern occurrence in coin flips

Abstract

We study a game in which one keeps flipping a coin until a given finite string of heads and tails occurs. We find the expected number of coin flips to end the game when the ending string consists of at most four maximal runs of heads or tails or alternates between heads and tails. This leads to some summation identities involving certain generalizations of the Fibonacci numbers.