an are defined by a1 = α, a2 = β, an+2 = anan+1/(2an - an+1). α, β are chosen so that an+1 ≠ 2an. For what α, β are infinitely many an integral?
Solution
A trivial induction shows that an+2 = αb/((n + 1)α - nβ) = αβ/(n(α - β) + α).
So we must have α = β, otherwise the denominator grows without limit. But if α = β, then all an = α. So infinitely many an are integral iff α = β = integer.
© John Scholes
jscholes@kalva.demon.co.uk
4 Dec 1999