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Jacobsthal representation numbers. (English) Zbl 0869.11013

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 \(\{j_n\}\).

MSC:

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