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a_{n+1}=(a_{n}^{2}+a_{n-1}^{2})/ a_{n-2} sequence of integer terms

Source: 3rd QEDMO 2006 p10

November 9, 2020

Sequencenumber theory with sequencesRecurrencenumber theory

Problem Statement

Define a sequence

\left( a_{n}\right) _{n\in\mathbb{N}}

a_{1}=a_{2}=a_{3}=1

and

a_{n+1}=\dfrac{a_{n}^{2}+a_{n-1}^{2}}{a_{n-2}}

for every integer

n\geq3

. Show that all elements

a_{i}

of this sequence are integers.
(L. J. Mordell and apparently Dana Scott, see also http://oeis.org/A064098)