Abstract
In this paper, we investigate the generalized Guglielmo sequences and we deal with, in detail, four special cases, namely, triangular, triangular-Lucas, oblong and pentagonal sequences. We present Binet's formulas, generating functions, Simson formulas, and the summation formulas for these sequences. Moreover, we give some identities and matrices related with these sequences.
Highlights
An oblong number On is a number which is the product of two consecutive integers, that is, a number of the form On = n(n + 1)
We present Simson’s formula of generalized Tribonacci numbers
Lemma 2 gives the following results as particular examples (generating functions of (r, s, t) and Lucas (r, s, t) numbers)
Summary
Note that oblong and triangular sequences have the following properties: Tn + Tn+1 = (n + 1), n n(n + 1)(n + 2). The purpose of this article is to generalize and investigate these interesting sequence of numbers (i.e., oblong, triangular and pentagonal numbers). In the case of two distinct roots, i.e., α = β = γ, for all integers n, Binet’s formula of (r, s, t) and Lucas (r, s, t) numbers (using initial conditions in (1.7)-(1.8)) can be expressed as follows: Theorem 4. (Single Root Case: α = β = γ) For all integers n, Binet’s formula of (r, s, t) and Lucas (r, s, t) numbers are. Lemma 2 gives the following results as particular examples (generating functions of (r, s, t) and Lucas (r, s, t) numbers).
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