1111111112334461111111-1-1-4-3-2-463Was war vor dem Startwert?

(1)

1 1 1 1 1

1

1 2

3 3

4 4 6

1 1

1

1 -1 -1 -4

-3

-2 -4

6

3

Was war vor dem Startwert?

(2)

Hans Walser

Was war vor dem Startwert?

Begabungsförderung Mathematik 30. März bis 1. April 2006, Erfurt

1 1 1 1 1

1 1

1 1 2

3 3

4 4 6 1

1 1 1 1

1 1 -1 -1 -4

-3 -2 -4

6 3

(3)

1 1

1 1 2

1 3 3 1

1 1 1

1

4 5

4 5 6

10 10 1

1 1

1 6

6

7 7

8 8

15 20 15 21 35 35 21 28 56 70 56 28

1

Hier sind Nullen.

a b a+b

Rekursion Startwert

(4)

1 1

1 1 2

1 3 3 1

1 1 1

1

4 5

4 5 6

10 10 1

1 1

1 6

6

7 7

8 8

15 20 15 21 35 35 21 28 56 70 56 28

1

Blaise Pascal 1623 - 1662

(5)

1 1

1 1 2

1 3 3

1 1 1

1

4 4

5 10 10

1 1

1 6

6

7 7

8 8

15 20 15 21 35 35 21 28 56 70 56 28

1

5

6

1

13

(6)

1 1 1 2 1

1 1 4

5 10 10

1 1

1 6

6

7 7

8 8

15 20 15 21 35 35 21 28 56 70 56 28 a b 1

a+b

Rekursion

1

5

6

1

1 13

1 1

3 4

3

58

(7)

1 1 1 2 1

1 1 4

5 10 10

1 1

1 6

6

7 7

8 8

15 20 15 21 35 35 21 28 56 70 56 28 a b 1

a+b

Rekursion

1

5

6

1

1 13

1 1

3 4

3

58

(8)

21 1

1 1

1 1 1 1

2 3

4 5

3 4

5 6

10 10 1

1 1

1 6

6

7 7

8 8

15 20 15 21 35 35 21 28 56 70 56 28

11

23 58

13 34 a b

a+b

Rekursion

(9)

21 1

1 1

1 1 1 1

2 3

4 5

3 4

5 6

10 10 1

1 1

1 6

6

7 7

8 8

15 20 15 21 35 35 21 28 56 70 56 28

11

23 58

13 34 a b

a+b

Rekursion

Leonardo Pisano Fibonacci um 1170 - 1250 f_n = f_n-1 + f_n-2

Rekursion

(10)

f_n = f_n-1 + f_n-2

n 1 2 3 4 5 6 ...

f_n 1 1 2 3 5 8 ...

Rekursion

(11)

f_n = f_n-1 + f_n-2

n 1 2 3 4 5 6 ...

f_n 1 1 2 3 5 8 ...

Rekursion

Startwerte

(12)

f_n = f_n-1 + f_n-2

n 1 2 3 4 5 6 ...

f_n 1 1 2 3 5 8 ...

Rekursion

Startwerte

(13)

f_n = f_n-1 + f_n-2

n 1 2 3 4 5 6 ...

f_n 1 1 2 3 5 8 ...

Rekursion

Was war vor den Startwerten ?

Startwerte

(14)

f_n = f_n-1 + f_n-2

n ... -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 ...

f_n 1 1 2 3 5 8 ...

Rekursion

Startwerte Was war vor den Startwerten ?

negative Zahlen

debiti

(15)

f_n = f_n-1 + f_n-2

n ... -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 ...

f_n 0 1 1 2 3 5 8 ...

Rekursion

Was war vor den Startwerten ? negative Zahlen

(16)

f_n = f_n-1 + f_n-2

n ... -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 ...

f_n 1 0 1 1 2 3 5 8 ...

Rekursion

(17)

f_n = f_n-1 + f_n-2

n ... -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 ...

f_n -1 1 0 1 1 2 3 5 8 ...

Rekursion

(18)

f_n = f_n-1 + f_n-2

n ... -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 ...

f_n 2 -1 1 0 1 1 2 3 5 8 ...

Rekursion

(19)

f_n = f_n-1 + f_n-2

n ... -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 ...

f_n -3 2 -1 1 0 1 1 2 3 5 8 ...

Rekursion

(20)

f_n = f_n-1 + f_n-2

n ... -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 ...

f_n ... -8 5 -3 2 -1 1 0 1 1 2 3 5 8 ...

Rekursion

Fibonacci-Folge mit

negativen Indizes

(21)

f_n = f_n-1 + f_n-2

n ... -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 ...

f_n ... -8 5 -3 2 -1 1 0 1 1 2 3 5 8 ...

Rekursion

Fibonacci-Folge mit

negativen Indizes

Geht das auch im Pascal-Dreieck?

(22)

1 1

1 1 1

2 3 4

3 4 6

1 1 1 1 1

–1 1

–2 –3 –4

3 –4

6

11 23

5 0

–11 –32 5

0 0 0 0

0 0 0

0 0

0

(23)

1 1

1 1 1

2 3 4

3 4 6

1 1 1 1 1

–1 1

–2 –3 –4

3 –4

6

11 23

5 0

–11 –32 5

0 0 0 0

0 0 0

0 0

0

a b a+b

Rekursion

(24)

a b a+b

Rekursion

1 1

1 1 1

2 3 4

3 4 6

1 1 1 1 1

–1 1

–2 –3 –4

3 –4

6

0 0 0 0 0

0 0 0 0

0 0 0

0 0 0 0

0 0 0

0 0

0

(25)

a b a+b

Rekursion Eigenbau?

(26)

a b a+b

Rekursion

1 1

2 3

4 1

1 1 1 1

–1 –2 –3 –4

0 0 0 0 0

0 0 0 0

0 0 0

0 0 0 0

Eigenbau?

(27)

a b a+b

Rekursion

1 1

2 3

4 1

1 1 1 1

–1 –2 –3 –4

0 0 0 0 0

0 0 0 0

0 0 0

0 0 0 0

Eigenbau?

2

(28)

a b a+b

Rekursion

1 1

2 3

4 1

1 1 1 1

–1 –2 –3 –4

0 0 0 0 0

0 0 0 0

0 0 0

0 0 0 0

Eigenbau?

2 4

7

(29)

a b a+b

Rekursion

1 1

2 3

4 1

1 1 1 1

–1 –2 –3 –4

0 0 0 0 0

0 0 0 0

0 0 0

0 0 0 0

Eigenbau?

2 4

7 1

1 2 4 7

(30)

a b a+b

Rekursion

1 1

2 3

4 1

1 1 1 1

–1 –2 –3 –4

0 0 0 0 0

0 0 0 0

0 0 0

0 0 0 0

Eigenbau?

2 4

7 1

1 2 4 7

8

(31)

a b a+b

Rekursion

1 1

2 3

4 1

1 1 1 1

–1 –2 –3 –4

0 0 0 0 0

0 0 0 0

0 0 0

0 0 0 0

Eigenbau?

2 4

7 1

1 2 4 7

8 10

14

(32)

a b a+b

Rekursion

1 1

2 3

4 1

1 1 1 1

–1 –2 –3 –4

0 0 0 0 0

0 0 0 0

0 0 0

0 0 0 0

Eigenbau?

2 4

7 1

1 2 4 7

8 10

14 7

6 4 0

(33)

a b a+b

Rekursion

1 1 1 1 1 1 1 1 1 1

2 2 2 2 2 2 2 2 2

4 4 4 4 4 4 4 4 4 4

8 8 8 8 8 8 8 8 8

16 16 16 16 16 16 16 16 16 16 32 32 32 32 32 32 32 32 32

1

2 1

4 1

8 1

16 1

16 1 16

(34)

a b a+b

Rekursion

1 1 1 1 1

–1 1

–2 –3 –4

3 –4

6

2 3

4

4 1

–1 –1 1 –1

–1 –1

–3 –6 0

0 0

0

0 0

0

0 0

0

0 0

0

0 0 0 0 0 0

0 0

0 0 0 0 0 0

0

0 0 0 0 0 0

0 0

0

(35)

a b a+b

Rekursion

1 1 1 1 1

–1 1

–2 –3 –4

3 –4

6

2 3

4

4 1

–1 –1 1 –1

–1 –1

–3 –6 0

0 0

0

0 0

0

0 0

0

0 0

0

0 0 0 0 0 0

(36)

a b a+b

Rekursion

1 1 1 1 1

–1 1

–2 –3 –4

3 –4

6

2 3

4

4 1

–1 –1 1 –1

–1 –1

–3 –6 0

0 0

0

0 0

0

0 0

0

0 0

0

0 0 0 0 0 0

1

-1 0 1 0 -1 1

-1 -1 0 1

(37)

a b a+b

Rekursion

1 1 1 1 1

–1 1

–2 –3 –4

3 –4

6

2 3

4

4 1

–1 –1 1 –1

–1 –1

–3 –6 0

0 0

0

0 0

0

0 0

0

0 0

0

0 0 0 0 0 0

1

-1 0 1 0 -1 1

-1 -1 0 1

n ... –5 –4 –3 –2 –1 0 1 2 3 4 5 6 ...

f_n ... –1 –1 0 1 1 0 –1 –1 0 1 1 0 ...

Periodische Folge, Rekursion: f_n = f_n-1 – f_n-2

(38)

1 1

1

–1 1 –1 1 –2

–3 –4

3 –4 6 0

0 0

0

0 0

0

0 0

0

0 0

0

0 0 0 0 0

0 0

0 0 0 0

0

0 0 0

0 0

1 1 1 1 1

–1 1

–2 –3 –4

3 –4

6

2 2 2 2 2

2 2

2 2 4

6 6

8 12 8 a b

a+b

Rekursion

(39)

a b a+b

Rekursion

1 1

1

–1 1 –1 1 –2

–3 –4

3 –4 6 0

0 0

0

0 0

0

0 0

0

0 0 0 0 0

0 0

0 0 0 0

0

0 0 0

0 0

1 1 1 1 1

–1 1

–2 –3 –4

3 –4

6

0

0 0

0

0 0 1 1 1 1 1

1 1

1 2

3 3

4 4

5 5

6 10 10

(40)

Binomische Formel

a + b

( )

ⁿ ⁼ _k

^∑

₌ⁿ₀

( )

ⁿn−k ^aⁿ⁻^k^b^k

a + b

( )

³ ⁼ _k

^∑

₌³₀

( )

33−k ^a³⁻^k^b^k ⁼ ^a³ ⁺ ^3a²^b ⁺ ^3ab² ⁺ ^b³

(41)

Binomische Formel

1

−4 1

6 −3 1

−4 3 −2 1

1 −1 1 −1 1

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

−5−5

( )

−5−4

( ) ( )

⁻⁴⁻⁴

−5−3

( ) ( )

⁻⁴⁻³

( )

⁻³⁻³

−5−2

( ) ( )

⁻⁴⁻²

( )

⁻³⁻²

( )

⁻²⁻²

−5−1

( ) ( )

⁻⁴⁻¹

( )

⁻³⁻¹

( )

⁻²⁻¹

( )

⁻¹⁻¹

00

( )

10

( ) ( )

¹¹

02

( ) ( )

¹²

( )

²²

03

( ) ( )

¹³

( )

²³

( )

³³

04

( ) ( )

¹⁴

( )

²⁴

( )

³⁴

( )

⁴⁴

(42)

1

−4 1

6 −3 1

−4 3 −2 1

1 −1 1 −1 1

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

−5−5

( )

−5−4

( ) ( )

⁻⁴⁻⁴

−5−3

( ) ( )

⁻⁴⁻³

( )

⁻³⁻³

−5−2

( ) ( )

⁻⁴⁻²

( )

⁻³⁻²

( )

⁻²⁻²

−5−1

( ) ( )

⁻⁴⁻¹

( )

⁻³⁻¹

( )

⁻²⁻¹

( )

⁻¹⁻¹

00

( )

10

( ) ( )

¹¹

02

( ) ( )

¹²

( )

²²

03

( ) ( )

¹³

( )

²³

( )

³³

04

( ) ( )

¹⁴

( )

²⁴

( )

³⁴

( )

⁴⁴

Binomische Formel

a + b

( )

³ ⁼ _k

^∑

₌³₀

( )

³3−k ^a³⁻^k^b^k ⁼ ^a³ ⁺ ^3a²^b ⁺^3ab² ⁺ ^b³

(43)

a +b

( )

⁻³ ⁼ _k

^∑

^∞₌₀

( )

−−33−k ^a⁻³⁻^k^b^k

= a⁻³ −3a⁻⁴b+6a⁻⁵b² −10a⁻⁶b³ +15a⁻⁷b⁴ −21a⁻⁸b⁵ ±!

1

−4 1

6 −3 1

−4 3 −2 1

1 −1 1 −1 1

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

−5−5

( )

−5−4

( ) ( )

⁻⁴⁻⁴

−5−3

( ) ( )

⁻⁴⁻³

( )

⁻³⁻³

−5−2

( ) ( )

⁻⁴⁻²

( )

⁻³⁻²

( )

⁻²⁻²

−5−1

( ) ( )

⁻⁴⁻¹

( )

⁻³⁻¹

( )

⁻²⁻¹

( )

⁻¹⁻¹

00

( )

10

( ) ( )

¹¹

02

( ) ( )

¹²

( )

²²

03

( ) ( )

¹³

( )

²³

( )

³³

04

( ) ( )

¹⁴

( )

²⁴

( )

³⁴

( )

⁴⁴

a + b

( )

³ ⁼ _k

^∑

₌³₀

( )

³3−k ^a³⁻^k^b^k ⁼ ^a³ ⁺ ^3a²^b ⁺^3ab² ⁺ ^b³

(44)

Sonderfall: a =1, b = x

f x

( )

⁼

( )

¹⁺ ^x ⁻³ ⁼¹⁻ ^3x ⁺ ⁶^x² ⁻¹⁰^x³ ⁺^15x⁴ ⁻ ^21x⁵ ^±^!

Brook Taylor 1685 - 1731

a +b

( )

⁻³ ⁼ _k

^∑

^∞₌₀

( )

−−33−k ^a⁻³⁻^k^b^k

= a⁻³ −3a⁻⁴b+6a⁻⁵b² −10a⁻⁶b³ +15a⁻⁷b⁴ −21a⁻⁸b⁵ ±!

(45)

f x

( )

⁼

( )

¹⁺ ^x ⁻³ ⁼¹⁻ ^3x ⁺ ⁶^x² ⁻¹⁰^x³ ⁺^15x⁴ ⁻ ^21x⁵ ^±^!

x

-2 -1 1 2

-1 1