Square classes and divisibility properties of Stern polynomials. (English) Zbl 1414.11040
Summary: The classical Stern sequence was extended by S. Klavžar et al. [Adv. Appl. Math. 39, No. 1, 86–95 (2007; Zbl 1171.11016)] to the Stern polynomials \(B_n(x)\) defined by \(B_0(x) = 0\), \(B_1(x) = 1, B_{2n}(x) = xB_n(x)\), and \(B_{2n+1}(x) = B_n(x) + B_{n+1}(x)\). In this paper we prove several divisibility results for these polynomials. We also find several infinite classes of positive integers n such that the Stern polynomials with index \(n^2\) are squares of polynomials which we give explicitly. We conjecture that, apart from two sporadic square Stern polynomials, we have characterized them all.
Keywords:
Stern polynomialsCitations:
Zbl 1171.11016Software:
OEISOnline Encyclopedia of Integer Sequences:
Jacobsthal sequence (or Jacobsthal numbers): a(n) = a(n-1) + 2*a(n-2), with a(0) = 0, a(1) = 1; also a(n) = nearest integer to 2^n/3.a(n) = a(n-1) + 2*a(n-2), a(0)=2, a(1)=3.
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