Formulas for Fibonacci-Like Sequences Produced by Pascal-Like Triangles

In this paper we are going to present three formulas to express Fibonacci-like sequences with the Fibonacci sequence. We constructed Pascal-like triangles using probabilities of a game, and these Pascal-like triangles can be considered generalizations of the well known Pascal\u27s triangle. Using these triangles, we can make Fibonacci-like sequences. We discovered an interesting formula for these Fibonacci-like sequences. We could generalize this formula, and got some interesting formulas that combine these Fibonacci-like sequences and the Fibonacci-sequence. These formulas can reveal very interesting relationships between Fibonacci-like sequences and the Fibonacci sequence, and we can expect a rich possibility of the research from these Fibonacci-like sequences