The Terms in Lucas Sequences Divisible by Their Indices

Terms in Elliptic Divisibility Sequences Divisible by Their Indices

Let D = (Dn)n≥1 be an elliptic divisibility sequence. We study the set S(D) of indices n satisfying n | Dn. In particular, given an index n ∈ S(D), we explain how to construct elements nd ∈ S(D), where d is either a prime divisor of Dn, or d is the product of the primes in an aliquot cycle for D. We also give bounds for the exceptional indices that are not constructed in this way.

Characterizing the Multiplicative Group of a Real Closed Field in Terms of its Divisible Maximal Subgroup

Linear Recurrence Sequences with Indices in Arithmetic Progression and Their Sums

For an arbitrary homogeneous linear recurrence sequence of order d with constant coe cients, we derive recurrence relations for all subsequences with indices in arithmetic progression. The coe cients of these recurrences are given explicitly in terms of partial Bell polynomials that depend on at most d 1 terms of the generalized Lucas sequence associated with the given recurrence. We also provi...

Determinants and permanents of Hessenberg matrices and generalized Lucas polynomials

In this paper, we give some determinantal and permanental representations of generalized Lucas polynomials, which are a general form of generalized bivariate Lucas p-polynomials, ordinary Lucas and Perrin sequences etc., by using various Hessenberg matrices. In addition, we show that determinant and permanent of these Hessenberg matrices can be obtained by using combinations. Then we show, the ...

Products of members of Lucas sequences with indices in an interval being a power

Let r and s be coprime nonzero integers with ∆ = r 2 + 4s = 0. Let α and β be the roots of the quadratic equation x 2 − rx − s = 0, and assume that α/β is not a root of 1. We make the convention that |α| ≥ |β|. Put (u n) n≥0 and (v n) n≥0 for the Lucas sequences of the first and second kind of roots α and β whose general terms are given by