We do what the title promises, and as a bonus, we get much simplified versions of these algorithms, that do not make any explicit mention of Gosper’s algorithm. © 2004 Elsevier Ltd. All rights reserved.

Maps with a denominator which vanishes in a subset of the phase space may generate unbounded trajectories which are not divergent, i.e. trajectories involving arbitrarily large values of the dynamic variables but which are not attracted to infinity. In this paper we propose some simple one-dimensional and two-dimensional recurrences which generate unbounded chaotic sequences, and through these ...

While in the univariate case solutions of linear recurrences with constant coe cients have rational generating functions we show that the multivariate case is much richer even though initial conditions have rational generating functions the corresponding solutions can have generating functions which are algebraic but not rational D nite but not algebraic and even non D nite Introduction The aim...

In this paper we consider the general sequences U„ and Vn satisfying the recurrences Un+2=mUn+l + Un, Vn+2=mV„+l+V„, (1.1) where m is a given positive integer, and UQ = 0, Ux = 1, V0 = 2, V1 = m. We shall occasionally refer to these sequences as U(m) and V(m) to emphasize their dependence on the parameter m. They can be represented by the Binet forms Un = {a-ni{a-P\ Vn = a+f3\ (1.2) where a+j3 ...

We resolve a conjecture proposed by D.E. Knuth concerning a recurrence arising in the satisfiability problem. Knuth’s recurrence resembles recurrences arising in the analysis of tries, in particular PATRICIA tries, and asymmetric leader election. We solve Knuth’s recurrence exactly and asymptotically, using analytic techniques such as the Mellin transform and analytic depoissonization.

We examine the parity of some statistics on lattice paths and Laguerre configurations, giving both algebraic and combinatorial treatments. For the former, we evaluate q-generating functions at q = −1; for the latter, we define appropriate parity-changing involutions on the associated structures. In addition, we furnish combinatorial proofs for a couple of related recurrences.

The moment-transfer approach is a standard tool for deriving limit laws of sequences of random variables satisfying a distributional recurrence. However, so far the approach could not be applied to certain “one-sided" recurrences with slowly varying moments and normal limit law. In this paper, we propose a modified version of the moment-transfer approach which can be applied to such recurrences...

In this paper, we analyse the dual pivot Quicksort, a variant of the standard Quicksort algorithm, in which two pivots are used for the partitioning of the array. We are solving recurrences of the expected number of key comparisons and exchanges performed by the algorithm, obtaining the exact and asymptotic total average values contributing to its time complexity. Further, we compute the averag...