AN ANALYSIS OF THE SEQUENCE Xn+2 = i m Xn+1 + Xn

We analyze the sequence Xn+2 = imXn+1 + Xn, with X1 = X2 = 1 + i, where i is the imaginary number and m is a real number. Plotting the sequence in the complex plane for different values of m, we see interesting figures from the conic sections. For values of m in the interval (−2, 2) we show that the figures generated are ellipses. We also provide analysis which prove that for certain values of m, the sequence generated is periodic with even period

Generalizing Random Fibonacci Sequences

We consider generalized Fibonacci sequences with recurrencerelation xn+p+1 = xn+p + xn, which have growth rates of the formlimn→∞ |xn|1/n that behave similarly to the golden ratio, (1 + √5)/2.Following Makover and McGowan’s analysis of the random Fibonacci se-quence, we find bounds for the value of E(|xn|)1/n for random sequencesgiven by xn+p+1 = ±xn+p + xn. Finally, we further generalize these ran-dom sequences using two parameters, p and q, and we experimentallyobserve how limn→∞ |xn|1/n contains surprising information about thedivisors of q +