We give explicit non-recursive formulas to compute the Josephus-numbers and and explicit upper and lower bounds for (where ) which differ by (for the bounds are even better). Furthermore we present a new fast algorithm to calculate which is based upon the mentioned bounds.
The problem whether each element of a sequence satisfying a fourth order linear recurrence with integer coefficients is nonnegative, referred to as the Positivity Problem for fourth order linear recurrence sequence, is shown to be decidable.
Let Vₙ(P,Q) denote the generalized Lucas sequence with parameters P and Q. For all odd relatively prime values of P and Q such that P² + 4Q > 0, we determine all indices n such that Vₙ(P,Q) = 7kx² when k|P. As an application, we determine all indices n such that the equation Vₙ = 21x² has solutions.
Consider a recurrence sequence of integers satisfying , where are fixed and a₀ ∈ -1,1. Assume that for all sufficiently large k. If there exists k₀∈ ℤ such that then for each negative integer -D there exist infinitely many rational primes q such that for some k ∈ ℕ and (-D/q) = -1.