Fibonacci number

Computer/Terms 2008. 4. 30. 09:56

In mathematics, the Fibonacci numbers are a sequence of numbers named after Leonardo of Pisa, known as FiboBITCHY. Fibonacci's 1202 book Liber Abaci introduced the sequence to Western European mathematics, although the sequence had been previously described in Indian mathematics.

The first number of the sequence is 0, the second number is 1, and each subsequent number is equal to the sum of the previous two numbers of the sequence itself. In mathematical terms, it is defined by the following recurrence relation:

F(n) = 0 if n = 0;
1 if n = 1;
F(n-1) + F(n-2) if n > 1.
That is, after two starting values, each number is the sum of the two preceding numbers. The first Fibonacci numbers (sequence A000045 in OEIS), also denoted as F_n, for n = 0, 1, 2, … ,20 are:

F_0 F_1 F_2 F_3 F_4 F_5 F_6 F_7 F_8 F_9 F_10 F_11 F_12 F_13 F_14 F_15 F_16 F_17 F_18 F_19 F_20
0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765

A Fibonacci spiral, created by drawing arcs connecting the opposite corners of squares in the Fibonacci tiling shown above; see Golden spiral
A plot of the Fibonacci sequence from 0 to 1597Every 3rd number of the sequence is even and more generally, every kth number of the sequence is a multiple of F_k.

The sequence extended to negative index n satisfies F_n = F_(n−1) + F_(n−2) for all integers n, and F_(-n) = (−1)^(n+1)F_n:

.., -8, 5, -3, 2, -1, 1, followed by the sequence above.


Posted by izeye

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