Research in Scientific Computing in Undergraduate Education

Integer values of recurrences

Victor Moll writes:

“The recurrence x[n] = (n + x[n - 1])/(1 - nx[n - 1]) comes from a simple finite sum of values of the arctangent function. Starting at x[0] = 0 you will see that x[n] \textrm{ is an integer for } n \leq 4. We have conjectured that this never happens again. The paper “Arithmetical properties of a sequence arising from an arctangent sum” (see below) contains some dynamical systems that needs to be explored.”

The aim of his project is to explore this conjecture computationally. In particular, what is the behavior of the sequence of fractional parts frac[x[n]]:=x[n]-\textrm{Floor}[x[n]] of the x[n] defined in the Amdeberhan et al (2007) article?

A plot of x[n] \textrm{ versus } n \textrm{ for } 0\leq n \leq 50 shows two values of n for which frac[x[n]] is small, namely n=16 \textrm{ and } n=37:

$latex x[n] \textrm{ versus} n \textrm{ for } 0\leq n \leq 50$

However, a check reveals that frac[x[16]]= \frac{3154072}{87995911}\approx 0.0358434 and frac[x[37]]=\frac{107145788085395791884}{62134508741384128482689} \approx 0.00172442 .

A plot of x[n] \textrm{ for } 0 \leq n \leq 10000 reveals spots, increasingly rare, where frac[x[n]]\approx 0:

References & readings

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