Jacobsthal representation numbers. (English) Zbl 0869.11013

Horadam, A. F.

Fibonacci Q. 34, No. 1, 40-54 (1996).

The author derives elementary properties of the sequences \[ J_{n+2} =J_{n+ 1} +2J_n,\;J_0=0,\;J_1=1,\;n\geq 0,\text{ and } j_{n+2} =j_{n+1} +2j_n,\;j_0= 2,\;j_1=1,\;n\geq 0, \] the Jacobsthal and Jacobsthal-Lucas sequences, respectively. He then shows that every positive integer \(N\) has a representation \(N=\sum_{j\geq 1} a_jJ_{n+1}\), where \(a_j\in \{0,1,2\}\) and \(a_j =2 \Rightarrow a_{j+1} =0\). A similar representation is also derived in terms of the sequence \(\{j_n\}\).

Reviewer: A.Knopfmacher (Wits)

MSC:

11B39	Fibonacci and Lucas numbers and polynomials and generalizations
11A67	Other number representations

Keywords:

Jacobsthal representation numbers; Jacobsthal sequences; Jacobsthal-Lucas sequences