Let ν(n) denote the number of 1’s in the binary representation of n. Properties of this function have been extensively studied in the literature due partly to its natural and frequent appearance in many concrete problems in diverse fields; see [16] and [42] the references therein. For more examples, see [1], [2], [5], [7], [8], [12], [20], [34]. The well-known Trollope-Delange formula (see [13]...

We improve an algorithm originally due to Chudnovsky and Chudnovsky for computing one selected term in a linear recurrent sequence with polynomial coefficients. Using baby-steps / giant-steps techniques, the nth term in such a sequence can be computed in time proportional to √ n, instead of n for a naive approach. As an intermediate result, we give a fast algorithm for computing the values take...

This paper examines the parallelization of a technique for speeding up the evaluation of potentially-complex real-valued functions at a large number of points. The technique being parallelixed generates a Chain of Recurrences (CR) which is then used to compute the function incrementally (i.e.) by using the results of one iteration in calculating the value of the function in the next iteration)....

In this paper several recurrences and formulas are presented leading to an upper bound and a lower bound, both logarithmic, for the expected height of a node in a heap. These bounds are of interest for algorithms that select the kth smallest element in a heap. CR categories: E.1, F.2.2, G.2.1.

The solutions to a change problem form restricted partitions. For one particular change problem, we look at the sequence representing the parity of these restricted partition values. It appears that the period of this sequence has not been studied. Through recurrences involving binomial coefficients, we find that the sequence has a period of 200.

For f(x) ∈ Z[x] and a ∈ Z, we let f(x) be the nth iterate of f(x), P (f, a) = {p prime : p|f(a) for some n}, and D(P (f, a)) denote the natural density of P (f, a) within the set of primes. A conjecture of Jones [5] indicates that D(P (f, a)) = 0 for most quadratic f . In this paper, we find an exceptional family of (f ,a) such that D(P (f, a)) > 0 by considering ft(x) = (x + t) − 2 − t and at ...

Le but de cette note est essentiellement de présenter des conjectures ainsi que des indications sur les raisons qui nous ont portées à énoncer ces conjectures. Les conjectures portent sur la forme explicite d’équations de récurrence que semble satisfaire la suite th(n), des nombres de tableaux de Young de hauteur bornée par h, ainsi que la suite t (2) h (n), des nombres de paires de tels tablea...

In this paper, we discuss the enumeration of words avoiding patterns with repeated letters. More specifically, we find recurrences (i.e. enumeration schemes) counting words avoiding any pattern of length 3 and words avoiding any monotone pattern.

Bousquet-Mélou and Petkovšek investigated the generating functions of multivariate linear recurrences with constant coefficients. We will give a reinterpretation of their theory by means of division theorems for formal power series, which clarifies the structural background and provides short, conceptual proofs. In addition, extending the division to the context of differential operators, the c...

Let tn be a sequence that satisfies a first order homogeneous recurrence tn = Q(n)tn−1, where Q ∈ Z[n]. The asymptotic behavior of the p-adic valuation of tn is described under the assumption that all the roots of Q in Z/pZ have nonvanishing derivative.