Pascal-like triangles and Fibonacci-like sequences

摘要

In [1] and [2] we demonstrated how Pascal-like triangles arose from theprobabilities associated with the various outcomes of a particular game (seeDefinition 1 below). It was also shown that they could be considered asgeneralisations of Pascal's triangle. In this article we show how Fibonacci-like sequences arise from our Pascal-like triangles, and demonstrate theexistence of simple relationships between these Fibonacci-like sequencesand the Fibonacci sequence itself. In addition we will investigate ageneralisation of the binomial coefficients that appears when considering anextended version of the game. We start by describing this game.