Tribonacci-like sequences and generalized Pascal's pyramids

Abstract

A well-known result of Feinberg and Shannon states that the tribonacci sequence can be detected by the so-called Pascal's pyramid. Here we will show that any tribonacci-like sequence can be obtained by the diagonals of the Feinberg's triangle associated to a suitable generalized Pascal's pyramid. The results also extend similar properties of Fibonacci-like sequence

An extension of Lucas identity via Pascal's triangle

“…(1.2) This sequence has many interesting properties (see [3,7,16,23] and references therein), in particular its 'Kepler limit', that is the limit of the ratio of consecutive terms (see [6,8]), is the so-called Plastic number. This number has a certain relevance in contemporary Architecture (see [12,16,17]), and it is considered the key to finding the right proportion among spatial figures at the same way that the Golden Mean is considered for the flat ones (see [15]).…”

“…It is known, even if not well-highlighted in literature (see [3,16,17,22]), that each term P m of the Padovan sequence can be described as sum of binomial coefficients. In fact, the following relation also known as Padovan identity holds:…”

“…To find numerical identities inside geometric configurations is a classical mathematical topic (see for example, [2,13,14,20] and references therein), so that it is natural to consider other diagonals with an increasing (k)-inclination (see Figure 2…”

On Third-Order $\overline{h}$-Jacobsthal and Third-Order $\overline{h}$-Jacobsthal--Lucas Sequences, and Related Quaternions

“…As the author had highlighted in [3], the numbers of Tribonacci type have many applications in distinct area of mathematics (see also [2] and reference therein). Therefore, it seems a natural question to investigate quaternions connected to either a Tribonacci-like sequence or more generally, to a whichever recursive sequence of third order; but it seems not easy to develop a general theory for quaternions connected to a whichever sequence too much different from Tribonacci sequence.…”

On ̄h-Jacobsthal and ̄h-Jacobsthal–Lucas sequences, and related quaternions

“…As the author had highlighted in [26], the numbers of Fibonacci type have many applications in distinct area of mathematics (see also [1], [28], [11] and reference therein). Therefore, it seems a natural question to investigate quaternions connected to either a Fibonacci-like sequence (see [28]) or more generally, to a whichever recursive sequence of second order; but it seems not easy to develop a general theory for quaternions connected to a whichever sequence too much different from Fibonacci or Lucas sequences.…”