What is the significance of the Fibonacci number?

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A Fibonacci number is a number from the Fibonacci sequence; this sequence is generated by starting with a(0)=1, a(1)=1 and then each nth term (n>1) is found by adding the proceeding two numbers. a(2)=a(0)+a(1)=1+1=2; a(3)=1+2=3 yielding the sequence 1,1,2,3,5,8,13,21,...

(1) In mathematics, the Fibonacci sequence is a special case of the Lucas numbers.

The ratio of adjacent Fibonacci numbers approaches the golden mean phi as the number of terms grows without bound.

(2) Historically the Fibonacci numbers were introduced in Europe in a book extolling the virtues of Hindu-Arabic numbers and arithmetic.

(3) Fibonacci numbers are said to appear in botany: in the ratio of angles of branches in certain plants (phyllotaxis) and in the number and arrangement in seeds in plants such as pine cones, sunflowers, and pineapples.

(4) The Fibonacci numbers are associated with spirals: in nature we see them in Nautilus shells and spiral galaxies.

(5) If we look at the ancestry of male honeybees we find the Fibonacci numbers --  a male has 1 parent, 2 grandparents, 3 great-grandparents, 5 great-great-grandparents, etc...

I hope this gives you some ideas to look into -- this sequence appears in zoology, botany, art, and in many areas of mathematics.

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