Abstract
There are many generalizations of the Fibonacci sequence such as the tribonacci, tetranacci, etc. A recent interesting generalization is the random analogue of the Fibonacci sequence defined by the random Fibonacci recurrence v(n) = +/- v(n-1) v(n-2), where v(1) = 1, v(2) = 1, and each +/- sign is independent and either + or - with probability 1/2. Viswanath shows that the absolute value of the n-th root of v(n) approaches C=1.1319882 as n tends to infinity. The constant C can be considered as the random analogue of the Golden ratio phi, which is the growth rate of the Fibonacci sequence. Although 9 shows up frequently in mathematics and nature, the growth rates of the other deterministic Fibonacci and the random Fibonacci sequences have not shown up as frequently. This observation prompted the investigation to see if the other constants can be found in nature or other systems. In this paper the presence of the generalized Fibonacci sequences and their related constants are shown in the geometry of the DNA, in the number of chromosomes in some animals and plants and in the measurements of the Arabian camel. An application of the random Fibonacci sequence in random integer generation is also briefly mentioned.