Abstract
Fibonacci-like polynomials, the roots of which are responsible for a cyclic behavior of orbits of a second-order two-parametric difference equation, are considered. Using Maple and Wolfram Alpha, the location of the largest and the smallest roots responsible for the cycles of period p among the roots responsible for the cycles of periods 2kp (period-doubling) and kp (period-multiplying) has been determined. These purely computational results of experimental mathematics, made possible by the use of modern digital tools, can be used as a motivation for confirmation through not-yet-developed methods of formal mathematics.
Highlights
IntroductionThe main results of this paper are purely computational, carried out in the contexts of
The main results of this paper are purely computational, carried out in the contexts ofMaple and Wolfram Alpha
Fibonacci-like polynomial P12 ( x ) that is responsible for a trivial cycle of period 14; It is the second root from the right on the ordered list of the roots of a 13th degree
Summary
The main results of this paper are purely computational, carried out in the contexts of. [1] have to be introduced To this end, one may recall that in Pascal’s triangle (Figure 1), which, according to ref. The second, third, fourth, and so on columns are the diagonals parallel, respectively, to the first one shifted by two rows down along the column on its immediate left. Through this process, the numbers on the shallow diagonals form rows of the rearranged Pascal’s triangle so that the sums of numbers in these rows are Fibonacci numbers (Figure 2).
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