Recurrence Formulas for Choulet Sequences

In this paper, we consider the so-called Choulet sequence an, n=0,1,2,3,… For any complex numbers a0,a1,k,l, it is defined by the formula an+1=∑j=0najan−j+k(n+1)+l for n=1,2,3,… We derive the recurrence formula of the form (n+1)an+∑j=15(uj+nvj)an−j=0 for n≥5, where uj,vj, j=1,…,5, are some constants that depend on a0,a1,k,l, but not on n. In the case when k=0, we obtain a shorter recurrence formula (n+1)an+∑j=14(uj+nvj)an−j=0 for n≥4. Both these formulas correspond to the recurrence formulas presented in the OEIS (The On-Line Encyclopedia of Integer Sequences) for many initial vectors (a0,a1,k,l)∈Z4. Most of them are listed as conjectures. So here we verify all of them using the same procedure by just inserting the values of a0,a1,k,l into the derived formulas. We also show that the Choulet sequence is a linear recurrence sequence if and only if (k,l)=(−a12,2a12+a1−2a0a1).