Integer values of recurrences
Victor Moll writes:
“The recurrence comes from a simple finite sum of values of the arctangent function. Starting at you will see that . 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 of the defined in the Amdeberhan et al (2007) article?
A plot of shows two values of for which is small, namely :However, a check reveals that and .
A plot of reveals spots, increasingly rare, where :
References & readings
- Tewodros Amdeberhan, Luis A. Medina & Victor H. Moll(2007) Arithmetical properties of a sequence arising from an arctangent sum. J. Number Theory (2007)(arithmetical _ properties _ of _a_ sequence _ arising _ from _ an _ arctangent _ sum)