Supppose there are integers a1,a2,dots and a polynomial p so that the integers p(a1),p(a2)... satisfy some linear recurrence, i.e. sump(ai)xi is a rational function of x. Must integers biinp−1(p(ai)) so that sumbixi is a rational function, necessarily exist?
(The answer is no if we ask for the function sumaixi to be rational, as can be seen when p(t)=t2 and ai being a random sequence of pm1)
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