Matrix Representation of the Fibonacci Numbers
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Matrix representation of the Fibonacci numbers
The Fibonacci sequence The pen name of Leonardo of Pisa was Fibonacci. Around 1200 AD Fibonacci posed the following problem: how would an isolated colony of rabbits grow from one pair of rabbits, assuming that each adult pair produced one mixed pair of rabbits each month, and each new pair became productive from the second month onwards? Solution In the first month there is just one pair of rabbits; this pair is reproductive, and produces a second pair of rabbits. In the second month, the original pair are reproductive and produce another (third) pair. The second pair does not produce offspring. In the third month, both the original and second pair are reproductive and produce offspring. The third pair does not produce offspring. And so forth. The following table indicates the number of reproductive and non-reproductive pairs at each stage. [Table goes here - download the original to see it.] At each stage the total number of pairs is the sum of the reproductive and non-reproductive pairs. The sequence of numbers 1, 2, 3, 5, 8, 13, …. is called the Fibonacci sequence. It is an example of a second-order recurrence relationship, in which the [Equation goes here - download the original to see it.] th term is defined in terms of the nth and [Equation goes here - download the original to see it.] th terms, thus [Equation goes here - download the original to see it.] where is the nth term. Also we have [Equation goes here - download the original to see it.] This is called a second-order recurrence relationship because the [Equation goes here - download the original to see it.] th term is defined by an operation on the two preceding terms.
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Contents of
Matrix Representation of the Fibonacci Numbers
1 Matrix representation of the Fibonacci numbers
2 General formula for the Fibonacci sequence
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