Generalized Pascal's triangles and associated k-Padovan-like sequences

One of the most interesting properties of Pascal's triangle is that the sequence of the sums of the elements on its diagonals is the best known recurrence sequence, the Fibonacci sequence. It is also known that other diagonals can be associated with other relevant recurrence sequences, such as the Padovan and k-Padovan sequences. In this paper, we see that similar properties also hold for diagonals of generalized Pascal's triangles. We show that the diagonal sums in generalized Pascal's triangles belong to the family of the so-called ‘k-Padovan-like sequences’ which are linear recurrences of order k with constant coefficients. A recurrence connection between the k-Padovan and k-Padovan-like sequences is derived.