Arithmetic functions with linear recurrence sequences

Exponential functions, linear recurrence sequences, and their applications

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...

Palindromes in Linear Recurrence Sequences

We prove that for any base b ≥ 2 and for any linear homogeneous recurrence sequence {an}n≥1 satisfying certain conditions, there exits a positive constant c > 0 such that #{n ≤ x : an is palindromic in base b} x1−c.

On Nearly Linear Recurrence Sequences

A nearly linear recurrence sequence (nlrs) is a complex sequence (an) with the property that there exist complex numbers A0,. . ., Ad−1 such that the sequence ( an+d + Ad−1an+d−1 + · · · + A0an )∞ n=0 is bounded. We give an asymptotic Binet-type formula for such sequences. We compare (an) with a natural linear recurrence sequence (lrs) (ãn) associated with it and prove under certain assumptions...

Zeros of linear recurrence sequences

Let f (x) = P0(x)α 0 + · · · + Pk(x)α k be an exponential polynomial over a field of zero characteristic. Assume that for each pair i, j with i 6= j , αi/αj is not a root of unity. Define 1 = ∑kj=0(deg Pj +1). We introduce a partition of {α0, . . . , αk} into subsets { αi0, . . . , αiki } (1 ≤ i ≤ m), which induces a decomposition of f into f = f1 +· · ·+fm, so that, for 1 ≤ i ≤ m, (αi0 : · · ·...