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Physical Sciences and Mathematics Commons™
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Full-Text Articles in Physical Sciences and Mathematics
Generalizing Random Fibonacci Sequences, Prashant Sansgiry, Ogul Duncan, David Duncan, Alexander Foster
Generalizing Random Fibonacci Sequences, Prashant Sansgiry, Ogul Duncan, David Duncan, Alexander Foster
Journal of the South Carolina Academy of Science
We consider generalized Fibonacci sequences with recurrence
relation xn+p+1 = xn+p + xn, which have growth rates of the form
limn→∞ |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 sequences
given by xn+p+1 = ±xn+p + xn. Finally, we further generalize these ran-
dom sequences using two parameters, p and q, and we experimentally
observe how limn→∞ |xn|1/n contains surprising information about the
divisors of q + 1
An Analysis Of The Sequence Xn+2 = I M Xn+1 + Xn, David Duncan, Prashant Sansgiry, Ogul Arslan, Jensen Meade
An Analysis Of The Sequence Xn+2 = I M Xn+1 + Xn, David Duncan, Prashant Sansgiry, Ogul Arslan, Jensen Meade
Journal of the South Carolina Academy of Science
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.