Hyperbolic Pascal triangles

摘要

In this paper, we introduce a new generalization of Pascal’s triangle. The new object is called the hyperbolic Pascal triangle since the mathematical background goes back to regular mosaics on the hyperbolic plane. We precisely describe the procedure of how to obtain a given type of hyperbolic Pascal triangle from a mosaic. Then we study certain quantitative properties such as the number, the sum, and the alternating sum of the elements of a row. Moreover, the pattern of the rows, and the appearance of some binary recurrences in a fixed hyperbolic triangle are investigated.