フィボナッチ数列

An+2=An+An+1
An+2=An+1+An
Fn+2=Fn+Fn+1
Fn+2=Fn+1+Fn
F0=0
F1=1
An=(1/√5){[(1+√5)/2]^2-[(1-√5)/2]^2}
An=[φ^n-(1-φ)^n]/√5
An=[φ^n-(-φ)^-n]/√5
Fn=(1/√5){[(1+√5)/2]^2-[(1-√5)/2]^2}
Fn=[φ^n-(1-φ)^n]/√5
Fn=[φ^n-(-φ)^-n]/√5

Fibonacci
Fibonacci Numbers
The Fibonacci numbers
Fibonacci Sequences
The Fibonacci sequences

フィボナッチ数
フィボナッチ数列

An+1/An=1.618
An/An+1=0.618
An+2/An=2.618
An/An+2=0.382
 

An+2/An+1=1.618

An+1/An=1.618

An/An-1=1.618

An-1/An=0.618
An/An+1=0.618

An+1/An+2=0.618

An+2/An=2.618

An+1/An-1=2.618

An/An-2=2.618

An/An+2=0.382

An-2/An=0.382


Fn+1/Fn=1.618
Fn/Fn+1=0.618
Fn+2/Fn=2.618
Fn/Fn+2=0.382

 

Fn+2/Fn+1=1.618

Fn+1/Fn=1.618

Fn/Fn-1=1.618

Fn-1/Fn=0.618
Fn/Fn+1=0.618

Fn+1/Fn+2=0.618

 

Fn+2/Fn=2.618

Fn+1/Fn-1=2.618

Fn/Fn-2=2.618

Fn/Fn+2=0.382

Fn-2/Fn=0.382


Fn+1/Fn=r=φ=1.6

Fn/Fn+1=1/r=1/φ=0.6

 

An/An-1=r=φ=1.6
An-1/An=1/r=1/φ=0.6
Fn/Fn-1=r=φ=1.6
Fn-1/Fn=1/r=1/φ=0.6

An+1/An=r=φ=1.6
An+1/An=1/r=1/φ=0.6
Fn+1/Fn=r=φ=1.6
Fn+1/Fn=1/r=1/φ=0.6

(1/φ)φ=1
φ=1.6
(1/1.6)1.6=(0.6)(1.6)
(0.6)(1.6)=0.96
φn=(Fn)φ+Fn-1
 

 

Geometric Progression
The geometric progressions
Geometric Sequence
The geometric sequences
等比数列
とうひすうれつ
Recurrence Relation
The recurrence relations
漸化式
ぜんかしき

 

Fibonacci

Fibonacci Numbers

The Fibonacci numbers

Fibonacci Sequences

The Fibonacci sequences

フィボナッチ数
フィボナッチ数列

xn=(n-2)x+(n-1)x

Fibonacci

Fibonacci Numbers

The Fibonacci numbers

Fibonacci Sequences

The Fibonacci sequences

フィボナッチ数
フィボナッチ数列

 

L1=lnT1

lnT1=S1

L1=S1

L2=lnT2

lnT2=S2

L2=S2

L3=L1+L2

L4=L2+L3

L5=L3+L4

Ln+2=Ln+Ln+1

L4=L2+(L1+L2)

L4=L1+2L2

L5=L3+L4

L5=(L1+L2)+(L1+2L2)

L5=2L1+3L2

Ln=Fn-2L1+Fn-1L2

Fibonacci Numbers

The Fibonacci numbers

Fibonacci Sequences

The Fibonacci sequences

フィボナッチ数
フィボナッチ数列

(Fn-2)(L1)+(Fn-1)(L2)=(Fn-2)(S1)+(Fn-1)(S2)

Fibonacci Numbers

The Fibonacci numbers

Fibonacci Sequences

The Fibonacci sequences

フィボナッチ数
フィボナッチ数列

(Ln)=(Fn-2)(S1)+(Fn-1)(S2)

Fibonacci Numbers

The Fibonacci numbers

Fibonacci Sequences

The Fibonacci sequences

フィボナッチ数
フィボナッチ数列

xn=(n-2)x+(n-1)x

Fibonacci Numbers

The Fibonacci numbers

Fibonacci Sequences

The Fibonacci sequences

フィボナッチ数
フィボナッチ数列

x^1=1x
x^2=1x+1
x^3=2x+1
x^4=3x+2
x^5=5x+3
x^6=8x+5
x^7=13x+8
x^8=21x+13
x^9=34x+21

黄金比
おうごんひ
Golden Triangle
The golden triangles
Goldenes Dreieck
Triangle D'or
黄金三角形
おうごんさんかっけい
Golden Rectangle
The golden rectangles
Goldenes Rechteck
黄金長方形
おうごんちょうほうけい
Pentagrammum
Pentagram
The pentagrams
Pentagramm
五芒星
ごぼうせい
Pentagonum
Pentagon
The pentagons
Fuenfeck
正五角形
せいごかっけい
Regular Dodecahedron
正十二面体
せいじゅうにめんたい

体温

x0=x

x1=1

x2=1+x=x0+x1

x3=1+(1+x)=x1+x2

x4=(1+x)+[1+(1+x)]=x2+x3

xn=x(n-2)+x(n-1)