This series begins with the third item, each item is equal to the sum of the first two items. Its general formula is (see figure) (also called "Binet.Alfred formula", which is an example of using irrational numbers to represent rational numbers. Interestingly, such a series of completely natural numbers, the general formula is actually expressed by irrational numbers.

Wonderful attributes

With the increase of the number of items in the series, the ratio of the previous item to the latter item approaches the golden section value of 0.6 180339887 ... From the second item, the square of each odd item is more than the product of the previous two items 1, and the square of each even item is less than the product of the previous two items 1. (Note: Odd and even terms refer to the parity of terms, not the parity of numbers in exponential columns. For example, the fourth item 3 is odd, but it is even, and the fifth item 5 is odd, and it is odd. If you think that the numbers 3 and 5 are odd terms, then you have misunderstood the meaning of the question and it doesn't make any sense. If you see such a topic: someone cut an 8*8 square into four pieces and put them together. In fact, this property of Fibonacci sequence is used: 5, 8, 13 are just the three adjacent terms in the sequence. In fact, the area difference between the front and rear blocks is indeed 1, but there is a slender slit in the back picture, which is not easy for ordinary people to notice. The nth term of Fibonacci sequence also means that the set {1, 2, ..., n} does not contain adjacent positive integers. Fibonacci sequence (f(n), f(0)=0, f( 1)= 1, f(2)= 1, other properties of f (3) = 2 ...):1.f. f(65438+= f(2n+ 1)- 1 4。 [f(0)]^2+[f( 1)]^2+…+[f(n)]^2=f(n]f(n+ 1)5.f(0)-f( 1)+f(2)-…+(- 1)^n f(n)=(- 1)n[f(n+ 1)-f(n)]+ 1 6。 f(m+n)= f(m- 1)f(n- 1)+f(m)f(。 7.[f(n)]^2=(- 1)^(n- 1)+f(n- 1]f(n+ 1)8.f(2n- 1)=[f(n)]^2-[f(n-2)]^2 9.3f(n)= f(n+2)+f(n-2) 10 . f(2n-2m-2) [f (2n)+f (2n+2)] = f (2m+2)+f (4n-2m) [n > m ≥-1,and n≥ 1] Fibonacci sequence