Continued fractions and linear recurrences
Continued fractions and linear recurrences
We prove that the numerators and denominators of the convergents to a real irrational number <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="theta"> <mml:semantics> <mml:mi>θ<!-- θ --></mml:mi> <mml:annotation encoding="application/x-tex">\theta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> satisfy a linear recurrence with constant coefficients if and only if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="theta"> <mml:semantics> <mml:mi>θ<!-- θ --></mml:mi> <mml:annotation …