フィボナッチ数列

フィボナッチ数列：漸化式 †

前の2項の和が次の項の値になるような数列です。 The nth Fibonacci number is the sum of the previous two Fibonacci numbers.

初期値　F(0)=0,F(1)=1
F(n) = F(n − 1) + F(n − 2)

• The first 21 Fibonacci numbers , also denoted as Fn, for n = 0, 1, 2, ... ,20 are:

  0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 

フィボナッチ数列：フィボナッチの兎の問題 †

In the West, the sequence was studied by Leonardo of Pisa, known as Fibonacci, in his Liber Abaci (1202). He considers the growth of an idealised (biologically unrealistic) rabbit population, assuming that:

In the "zeroth" month, there is one pair of rabbits (additional pairs of rabbits = 0). In the first month, the first pair begets another pair (additional pairs of rabbits = 1). In the second month, both pairs of rabbits have another pair, and the first pair dies (additional pairs of rabbits = 1). In the third month, the second pair and the new two pairs have a total of three new pairs, and the older second pair dies (additional pairs of rabbits = 2). The laws of this are that each pair of rabbits has 2 pairs in its lifetime, and dies.

Let the population at month n be F(n). At this time, only rabbits who were alive at month n − 2 are fertile and produce offspring, so F(n − 2) pairs are added to the current population of F(n − 1). Thus the total is F(n) = F(n − 1) + F(n − 2).

黄金数との関係 †

フィボナッチ数Fnのｎを大きくしていくと、続く2項の比　Fn+1/Fn　は、黄金数に限りなく近づく。

Limit　n--->∞　Fn+1/Fn　=　φ　=　(1+√5)/2　=1.61803

• Johannes Kepler observed that the ratio of consecutive Fibonacci numbers converges. He wrote that "as 5 is to 8 so is 8 to 13, practically, and as 8 is to 13, so is 13 to 21 almost”, and concluded that the limit approaches the golden ratio.

Fnは黄金数φで表わされる

Fn = [φ^n　-(1-φ)^n　]/√5