# Article

Full entry | PDF   (0.3 MB)
Keywords:
sequences of binary polynomials; Stern-Brocot sequence; perfect fields of characteristic 2
Summary:
In this paper, we study the properties of the sequence of polynomials given by $g_0=0,~g_1=1$, $g_{n+1}=g_n+\Delta g_{n-1}$ for $n\ge 1$, where $\Delta \in {\mathbb F}_q[t]$ is non-constant and the characteristic of ${\mathbb F}_q$ is $2$. This complements some results from R. Euler, L.H. Gallardo: On explicit formulae and linear recurrent sequences, Acta Math. Univ. Comenianae, 80 (2011) 213-219.
References:
[1] Cherly, J., Gallardo, L., Vaserstein, L., Wheland, E.: Solving quadratic equations over polynomial rings of characteristic two. Publ. Math., 42, 1998, 131-142, DOI 10.5565/PUBLMAT_42198_06 | MR 1628154 | Zbl 0915.13017
[2] Euler, R., Gallardo, L.H.: On explicit formulae and linear recurrent sequences. Acta Math. Univ. Comenianae, 80, 2011, 213-219, MR 2835276 | Zbl 1255.11061
[3] He, T.-X., Shiue, P.J.-S.: On sequences of numbers and polynomials defined by linear recurrence relations of order 2. Int. J. Math. Math, Sci., 2009, Art. ID 709386, 21 pp.. MR 2552555 | Zbl 1193.11014
[4] Northshield, S.: Stern's diatomic sequence $0,1,1,2,1,3,2,3,1,4,...$. Amer. Math. Monthly, 117, 2010, 581-598, DOI 10.4169/000298910X496714 | MR 2681519 | Zbl 1210.11035
[5] Sloane, N.J.A.: OEIS. https://oeis.org/

Partner of