MATHS IN NATURE AND ARTS FIBONACCI’S SEQUENCE AND GOLDEN RATIO.
Fibonacci Numbers - Bard Math Circle · 11/12/2019 · Fibonacci Numbers Japheth Wood, PhD...
Transcript of Fibonacci Numbers - Bard Math Circle · 11/12/2019 · Fibonacci Numbers Japheth Wood, PhD...
![Page 1: Fibonacci Numbers - Bard Math Circle · 11/12/2019 · Fibonacci Numbers Japheth Wood, PhD Fibonacci Book Rabbit Problem Some problems to solve Solutions Binet’s Formula Fibonacci’s](https://reader034.fdocuments.us/reader034/viewer/2022051410/60288b3a40297e1cab780ce5/html5/thumbnails/1.jpg)
FibonacciNumbers
Japheth Wood,PhD
Fibonacci BookRabbit Problem
Some problems tosolve
Solutions
Binet’s Formula
Fibonacci Numbers
Japheth Wood, PhD
Bard Math Circle AMC 8November 12, 2019
![Page 2: Fibonacci Numbers - Bard Math Circle · 11/12/2019 · Fibonacci Numbers Japheth Wood, PhD Fibonacci Book Rabbit Problem Some problems to solve Solutions Binet’s Formula Fibonacci’s](https://reader034.fdocuments.us/reader034/viewer/2022051410/60288b3a40297e1cab780ce5/html5/thumbnails/2.jpg)
FibonacciNumbers
Japheth Wood,PhD
Fibonacci BookRabbit Problem
Some problems tosolve
Solutions
Binet’s Formula
Fibonacci’s Rabbit Problem
“A certain man put a pair of rabbits ina place surrounded on all sides by a wall.How many pairs of rabbits can be pro-duced from that pair in a year if it is sup-posed that every month each pair begetsa new pair which from the second monthon becomes productive?”
—A problem from the third section of Liber abaci (1202).
(https://www-history.mcs.st-andrews.ac.uk/Biographies/Fibonacci.html)
Recursive Definition
F(Next) = F(Current) + F(Productive), F(0) = F(1) = 1
Rabbit Population
Month: 0 1
2 3 4 5 6 7 8 9 10
Rabbits: 1 1 2 3 5 8 13 21 34 55 89
![Page 3: Fibonacci Numbers - Bard Math Circle · 11/12/2019 · Fibonacci Numbers Japheth Wood, PhD Fibonacci Book Rabbit Problem Some problems to solve Solutions Binet’s Formula Fibonacci’s](https://reader034.fdocuments.us/reader034/viewer/2022051410/60288b3a40297e1cab780ce5/html5/thumbnails/3.jpg)
FibonacciNumbers
Japheth Wood,PhD
Fibonacci BookRabbit Problem
Some problems tosolve
Solutions
Binet’s Formula
Fibonacci’s Rabbit Problem
“A certain man put a pair of rabbits ina place surrounded on all sides by a wall.How many pairs of rabbits can be pro-duced from that pair in a year if it is sup-posed that every month each pair begetsa new pair which from the second monthon becomes productive?”
—A problem from the third section of Liber abaci (1202).
(https://www-history.mcs.st-andrews.ac.uk/Biographies/Fibonacci.html)
Recursive Definition
F(Next) = F(Current) + F(Productive), F(0) = F(1) = 1
Rabbit Population
Month: 0 1
2 3 4 5 6 7 8 9 10
Rabbits: 1 1 2 3 5 8 13 21 34 55 89
![Page 4: Fibonacci Numbers - Bard Math Circle · 11/12/2019 · Fibonacci Numbers Japheth Wood, PhD Fibonacci Book Rabbit Problem Some problems to solve Solutions Binet’s Formula Fibonacci’s](https://reader034.fdocuments.us/reader034/viewer/2022051410/60288b3a40297e1cab780ce5/html5/thumbnails/4.jpg)
FibonacciNumbers
Japheth Wood,PhD
Fibonacci BookRabbit Problem
Some problems tosolve
Solutions
Binet’s Formula
Fibonacci’s Rabbit Problem
“A certain man put a pair of rabbits ina place surrounded on all sides by a wall.How many pairs of rabbits can be pro-duced from that pair in a year if it is sup-posed that every month each pair begetsa new pair which from the second monthon becomes productive?”
—A problem from the third section of Liber abaci (1202).
(https://www-history.mcs.st-andrews.ac.uk/Biographies/Fibonacci.html)
Recursive Definition
F(Next) = F(Current) + F(Productive), F(0) = F(1) = 1
Rabbit Population
Month: 0 1 2 3 4 5 6 7 8 9 10
Rabbits: 1 1
2 3 5 8 13 21 34 55 89
![Page 5: Fibonacci Numbers - Bard Math Circle · 11/12/2019 · Fibonacci Numbers Japheth Wood, PhD Fibonacci Book Rabbit Problem Some problems to solve Solutions Binet’s Formula Fibonacci’s](https://reader034.fdocuments.us/reader034/viewer/2022051410/60288b3a40297e1cab780ce5/html5/thumbnails/5.jpg)
FibonacciNumbers
Japheth Wood,PhD
Fibonacci BookRabbit Problem
Some problems tosolve
Solutions
Binet’s Formula
Fibonacci’s Rabbit Problem
“A certain man put a pair of rabbits ina place surrounded on all sides by a wall.How many pairs of rabbits can be pro-duced from that pair in a year if it is sup-posed that every month each pair begetsa new pair which from the second monthon becomes productive?”
—A problem from the third section of Liber abaci (1202).
(https://www-history.mcs.st-andrews.ac.uk/Biographies/Fibonacci.html)
Recursive Definition
F(Next) = F(Current) + F(Productive), F(0) = F(1) = 1
Rabbit Population
Month: 0 1 2 3 4 5 6 7 8 9 10
Rabbits: 1 1 2
3 5 8 13 21 34 55 89
![Page 6: Fibonacci Numbers - Bard Math Circle · 11/12/2019 · Fibonacci Numbers Japheth Wood, PhD Fibonacci Book Rabbit Problem Some problems to solve Solutions Binet’s Formula Fibonacci’s](https://reader034.fdocuments.us/reader034/viewer/2022051410/60288b3a40297e1cab780ce5/html5/thumbnails/6.jpg)
FibonacciNumbers
Japheth Wood,PhD
Fibonacci BookRabbit Problem
Some problems tosolve
Solutions
Binet’s Formula
Fibonacci’s Rabbit Problem
“A certain man put a pair of rabbits ina place surrounded on all sides by a wall.How many pairs of rabbits can be pro-duced from that pair in a year if it is sup-posed that every month each pair begetsa new pair which from the second monthon becomes productive?”
—A problem from the third section of Liber abaci (1202).
(https://www-history.mcs.st-andrews.ac.uk/Biographies/Fibonacci.html)
Recursive Definition
F(Next) = F(Current) + F(Productive), F(0) = F(1) = 1
Rabbit Population
Month: 0 1 2 3 4 5 6 7 8 9 10
Rabbits: 1 1 2 3
5 8 13 21 34 55 89
![Page 7: Fibonacci Numbers - Bard Math Circle · 11/12/2019 · Fibonacci Numbers Japheth Wood, PhD Fibonacci Book Rabbit Problem Some problems to solve Solutions Binet’s Formula Fibonacci’s](https://reader034.fdocuments.us/reader034/viewer/2022051410/60288b3a40297e1cab780ce5/html5/thumbnails/7.jpg)
FibonacciNumbers
Japheth Wood,PhD
Fibonacci BookRabbit Problem
Some problems tosolve
Solutions
Binet’s Formula
Fibonacci’s Rabbit Problem
“A certain man put a pair of rabbits ina place surrounded on all sides by a wall.How many pairs of rabbits can be pro-duced from that pair in a year if it is sup-posed that every month each pair begetsa new pair which from the second monthon becomes productive?”
—A problem from the third section of Liber abaci (1202).
(https://www-history.mcs.st-andrews.ac.uk/Biographies/Fibonacci.html)
Recursive Definition
F(Next) = F(Current) + F(Productive), F(0) = F(1) = 1
Rabbit Population
Month: 0 1 2 3 4 5 6 7 8 9 10
Rabbits: 1 1 2 3 5
8 13 21 34 55 89
![Page 8: Fibonacci Numbers - Bard Math Circle · 11/12/2019 · Fibonacci Numbers Japheth Wood, PhD Fibonacci Book Rabbit Problem Some problems to solve Solutions Binet’s Formula Fibonacci’s](https://reader034.fdocuments.us/reader034/viewer/2022051410/60288b3a40297e1cab780ce5/html5/thumbnails/8.jpg)
FibonacciNumbers
Japheth Wood,PhD
Fibonacci BookRabbit Problem
Some problems tosolve
Solutions
Binet’s Formula
Fibonacci’s Rabbit Problem
“A certain man put a pair of rabbits ina place surrounded on all sides by a wall.How many pairs of rabbits can be pro-duced from that pair in a year if it is sup-posed that every month each pair begetsa new pair which from the second monthon becomes productive?”
—A problem from the third section of Liber abaci (1202).
(https://www-history.mcs.st-andrews.ac.uk/Biographies/Fibonacci.html)
Recursive Definition
F(Next) = F(Current) + F(Productive), F(0) = F(1) = 1
Rabbit Population
Month: 0 1 2 3 4 5 6 7 8 9 10
Rabbits: 1 1 2 3 5 8
13 21 34 55 89
![Page 9: Fibonacci Numbers - Bard Math Circle · 11/12/2019 · Fibonacci Numbers Japheth Wood, PhD Fibonacci Book Rabbit Problem Some problems to solve Solutions Binet’s Formula Fibonacci’s](https://reader034.fdocuments.us/reader034/viewer/2022051410/60288b3a40297e1cab780ce5/html5/thumbnails/9.jpg)
FibonacciNumbers
Japheth Wood,PhD
Fibonacci BookRabbit Problem
Some problems tosolve
Solutions
Binet’s Formula
Fibonacci’s Rabbit Problem
“A certain man put a pair of rabbits ina place surrounded on all sides by a wall.How many pairs of rabbits can be pro-duced from that pair in a year if it is sup-posed that every month each pair begetsa new pair which from the second monthon becomes productive?”
—A problem from the third section of Liber abaci (1202).
(https://www-history.mcs.st-andrews.ac.uk/Biographies/Fibonacci.html)
Recursive Definition
F(Next) = F(Current) + F(Productive), F(0) = F(1) = 1
Rabbit Population
Month: 0 1 2 3 4 5 6 7 8 9 10
Rabbits: 1 1 2 3 5 8 13
21 34 55 89
![Page 10: Fibonacci Numbers - Bard Math Circle · 11/12/2019 · Fibonacci Numbers Japheth Wood, PhD Fibonacci Book Rabbit Problem Some problems to solve Solutions Binet’s Formula Fibonacci’s](https://reader034.fdocuments.us/reader034/viewer/2022051410/60288b3a40297e1cab780ce5/html5/thumbnails/10.jpg)
FibonacciNumbers
Japheth Wood,PhD
Fibonacci BookRabbit Problem
Some problems tosolve
Solutions
Binet’s Formula
Fibonacci’s Rabbit Problem
“A certain man put a pair of rabbits ina place surrounded on all sides by a wall.How many pairs of rabbits can be pro-duced from that pair in a year if it is sup-posed that every month each pair begetsa new pair which from the second monthon becomes productive?”
—A problem from the third section of Liber abaci (1202).
(https://www-history.mcs.st-andrews.ac.uk/Biographies/Fibonacci.html)
Recursive Definition
F(Next) = F(Current) + F(Productive), F(0) = F(1) = 1
Rabbit Population
Month: 0 1 2 3 4 5 6 7 8 9 10
Rabbits: 1 1 2 3 5 8 13 21 34 55 89
![Page 11: Fibonacci Numbers - Bard Math Circle · 11/12/2019 · Fibonacci Numbers Japheth Wood, PhD Fibonacci Book Rabbit Problem Some problems to solve Solutions Binet’s Formula Fibonacci’s](https://reader034.fdocuments.us/reader034/viewer/2022051410/60288b3a40297e1cab780ce5/html5/thumbnails/11.jpg)
FibonacciNumbers
Japheth Wood,PhD
Fibonacci BookRabbit Problem
Some problems tosolve
Solutions
Binet’s Formula
Solve one of these problems:
A composition of n is a wayto write n as the sum of posi-tive integers (order matters).How many compositions arethere of 8 that don’t use 1?
How many increasing pathsare there through the
honeycomb from 1 to 7?
How many compositions arethere of 7 into odd parts?
How many ways are there toclimb a set of 6 stairs, one ortwo steps at a time?
How many subsets are thereof {1, 2, 3, 4, 5} that includeno two consecutive numbers?
How many binary sequencesof length 5 are there, with no
consecutive 0’s?
In how many ways can youtile a 2× 6 rectanglewith 2× 1 dominoes?
Find 6 positive integersolutions (x , y) ofy 2 − xy − x2 = ±1.
![Page 12: Fibonacci Numbers - Bard Math Circle · 11/12/2019 · Fibonacci Numbers Japheth Wood, PhD Fibonacci Book Rabbit Problem Some problems to solve Solutions Binet’s Formula Fibonacci’s](https://reader034.fdocuments.us/reader034/viewer/2022051410/60288b3a40297e1cab780ce5/html5/thumbnails/12.jpg)
FibonacciNumbers
Japheth Wood,PhD
Fibonacci BookRabbit Problem
Some problems tosolve
Solutions
Binet’s Formula
How many increasing paths are therethrough the honeycomb from 1 to 7?
· · · Path(5)+Path(6)=Path(7)Path(n − 2) + Path(n − 1) = Path(n)
n 1 2 3 4 5 6 7
P(n) 1 1 2 3 5 8 13
Paths ending 5-7 Paths ending 6-7
![Page 13: Fibonacci Numbers - Bard Math Circle · 11/12/2019 · Fibonacci Numbers Japheth Wood, PhD Fibonacci Book Rabbit Problem Some problems to solve Solutions Binet’s Formula Fibonacci’s](https://reader034.fdocuments.us/reader034/viewer/2022051410/60288b3a40297e1cab780ce5/html5/thumbnails/13.jpg)
FibonacciNumbers
Japheth Wood,PhD
Fibonacci BookRabbit Problem
Some problems tosolve
Solutions
Binet’s Formula
How many increasing paths are therethrough the honeycomb from 1 to 7?
· · · Path(5)+Path(6)=
Path(7)
Path(n − 2) + Path(n − 1) = Path(n)
n 1 2 3 4 5 6 7
P(n) 1 1 2 3 5 8 13
Paths ending 5-7 Paths ending 6-7
![Page 14: Fibonacci Numbers - Bard Math Circle · 11/12/2019 · Fibonacci Numbers Japheth Wood, PhD Fibonacci Book Rabbit Problem Some problems to solve Solutions Binet’s Formula Fibonacci’s](https://reader034.fdocuments.us/reader034/viewer/2022051410/60288b3a40297e1cab780ce5/html5/thumbnails/14.jpg)
FibonacciNumbers
Japheth Wood,PhD
Fibonacci BookRabbit Problem
Some problems tosolve
Solutions
Binet’s Formula
How many increasing paths are therethrough the honeycomb from 1 to 7?
· · · Path(5)+
Path(6)
=
Path(7)
Path(n − 2) + Path(n − 1) = Path(n)
n 1 2 3 4 5 6 7
P(n) 1 1 2 3 5 8 13
Paths ending 5-7 Paths ending 6-7
![Page 15: Fibonacci Numbers - Bard Math Circle · 11/12/2019 · Fibonacci Numbers Japheth Wood, PhD Fibonacci Book Rabbit Problem Some problems to solve Solutions Binet’s Formula Fibonacci’s](https://reader034.fdocuments.us/reader034/viewer/2022051410/60288b3a40297e1cab780ce5/html5/thumbnails/15.jpg)
FibonacciNumbers
Japheth Wood,PhD
Fibonacci BookRabbit Problem
Some problems tosolve
Solutions
Binet’s Formula
How many increasing paths are therethrough the honeycomb from 1 to 7?
· · ·
Path(5)
+
Path(6)
=
Path(7)
Path(n − 2) + Path(n − 1) = Path(n)
n 1 2 3 4 5 6 7
P(n) 1 1 2 3 5 8 13
Paths ending 5-7 Paths ending 6-7
![Page 16: Fibonacci Numbers - Bard Math Circle · 11/12/2019 · Fibonacci Numbers Japheth Wood, PhD Fibonacci Book Rabbit Problem Some problems to solve Solutions Binet’s Formula Fibonacci’s](https://reader034.fdocuments.us/reader034/viewer/2022051410/60288b3a40297e1cab780ce5/html5/thumbnails/16.jpg)
FibonacciNumbers
Japheth Wood,PhD
Fibonacci BookRabbit Problem
Some problems tosolve
Solutions
Binet’s Formula
How many increasing paths are therethrough the honeycomb from 1 to 7?
· · · Path(5)
+
Path(6)
=
Path(7)
Path(n − 2) + Path(n − 1) = Path(n)
n 1 2 3 4 5 6 7
P(n) 1 1 2 3 5 8 13
Paths ending 5-7 Paths ending 6-7
![Page 17: Fibonacci Numbers - Bard Math Circle · 11/12/2019 · Fibonacci Numbers Japheth Wood, PhD Fibonacci Book Rabbit Problem Some problems to solve Solutions Binet’s Formula Fibonacci’s](https://reader034.fdocuments.us/reader034/viewer/2022051410/60288b3a40297e1cab780ce5/html5/thumbnails/17.jpg)
FibonacciNumbers
Japheth Wood,PhD
Fibonacci BookRabbit Problem
Some problems tosolve
Solutions
Binet’s Formula
How many increasing paths are therethrough the honeycomb from 1 to 7?
· · · Path(5)
+
Path(6)
=
Path(7)
Path(n − 2) + Path(n − 1) =
Path(n)
n 1 2 3 4 5 6 7
P(n) 1 1 2 3 5 8 13
Paths ending 5-7 Paths ending 6-7
![Page 18: Fibonacci Numbers - Bard Math Circle · 11/12/2019 · Fibonacci Numbers Japheth Wood, PhD Fibonacci Book Rabbit Problem Some problems to solve Solutions Binet’s Formula Fibonacci’s](https://reader034.fdocuments.us/reader034/viewer/2022051410/60288b3a40297e1cab780ce5/html5/thumbnails/18.jpg)
FibonacciNumbers
Japheth Wood,PhD
Fibonacci BookRabbit Problem
Some problems tosolve
Solutions
Binet’s Formula
How many increasing paths are therethrough the honeycomb from 1 to 7?
· · · Path(5)
+
Path(6)
=
Path(7)
Path(n − 2) + Path(n − 1) =
Path(n)n 1 2 3 4 5 6 7
P(n)
1 1 2 3 5 8 13
Paths ending 5-7 Paths ending 6-7
![Page 19: Fibonacci Numbers - Bard Math Circle · 11/12/2019 · Fibonacci Numbers Japheth Wood, PhD Fibonacci Book Rabbit Problem Some problems to solve Solutions Binet’s Formula Fibonacci’s](https://reader034.fdocuments.us/reader034/viewer/2022051410/60288b3a40297e1cab780ce5/html5/thumbnails/19.jpg)
FibonacciNumbers
Japheth Wood,PhD
Fibonacci BookRabbit Problem
Some problems tosolve
Solutions
Binet’s Formula
How many increasing paths are therethrough the honeycomb from 1 to 7?
· · · Path(5)
+
Path(6)
=
Path(7)
Path(n − 2) + Path(n − 1) =
Path(n)n 1 2 3 4 5 6 7
P(n)
1 1 2 3 5 8 13
Paths ending 5-7
Paths ending 6-7
![Page 20: Fibonacci Numbers - Bard Math Circle · 11/12/2019 · Fibonacci Numbers Japheth Wood, PhD Fibonacci Book Rabbit Problem Some problems to solve Solutions Binet’s Formula Fibonacci’s](https://reader034.fdocuments.us/reader034/viewer/2022051410/60288b3a40297e1cab780ce5/html5/thumbnails/20.jpg)
FibonacciNumbers
Japheth Wood,PhD
Fibonacci BookRabbit Problem
Some problems tosolve
Solutions
Binet’s Formula
How many increasing paths are therethrough the honeycomb from 1 to 7?
· · · Path(5)
+
Path(6)
=
Path(7)
Path(n − 2) + Path(n − 1) =
Path(n)n 1 2 3 4 5 6 7
P(n)
1 1 2 3 5 8 13
Paths ending 5-7 Paths ending 6-7
![Page 21: Fibonacci Numbers - Bard Math Circle · 11/12/2019 · Fibonacci Numbers Japheth Wood, PhD Fibonacci Book Rabbit Problem Some problems to solve Solutions Binet’s Formula Fibonacci’s](https://reader034.fdocuments.us/reader034/viewer/2022051410/60288b3a40297e1cab780ce5/html5/thumbnails/21.jpg)
FibonacciNumbers
Japheth Wood,PhD
Fibonacci BookRabbit Problem
Some problems tosolve
Solutions
Binet’s Formula
How many increasing paths are therethrough the honeycomb from 1 to 7?
· · ·
Path(5)+Path(6)=Path(7)
Path(n − 2) + Path(n − 1) =
Path(n)n 1 2 3 4 5 6 7
P(n)
1 1 2 3 5 8 13
Paths ending 5-7 Paths ending 6-7
![Page 22: Fibonacci Numbers - Bard Math Circle · 11/12/2019 · Fibonacci Numbers Japheth Wood, PhD Fibonacci Book Rabbit Problem Some problems to solve Solutions Binet’s Formula Fibonacci’s](https://reader034.fdocuments.us/reader034/viewer/2022051410/60288b3a40297e1cab780ce5/html5/thumbnails/22.jpg)
FibonacciNumbers
Japheth Wood,PhD
Fibonacci BookRabbit Problem
Some problems tosolve
Solutions
Binet’s Formula
How many increasing paths are therethrough the honeycomb from 1 to 7?
· · ·
Path(5)+Path(6)=Path(7)Path(n − 2) + Path(n − 1) = Path(n)
n 1 2 3 4 5 6 7P(n)
1 1 2 3 5 8 13
Paths ending 5-7 Paths ending 6-7
![Page 23: Fibonacci Numbers - Bard Math Circle · 11/12/2019 · Fibonacci Numbers Japheth Wood, PhD Fibonacci Book Rabbit Problem Some problems to solve Solutions Binet’s Formula Fibonacci’s](https://reader034.fdocuments.us/reader034/viewer/2022051410/60288b3a40297e1cab780ce5/html5/thumbnails/23.jpg)
FibonacciNumbers
Japheth Wood,PhD
Fibonacci BookRabbit Problem
Some problems tosolve
Solutions
Binet’s Formula
How many increasing paths are therethrough the honeycomb from 1 to 7?
· · ·
Path(5)+Path(6)=Path(7)Path(n − 2) + Path(n − 1) = Path(n)
n 1 2 3 4 5 6 7P(n) 1 1
2 3 5 8 13
Paths ending 5-7 Paths ending 6-7
![Page 24: Fibonacci Numbers - Bard Math Circle · 11/12/2019 · Fibonacci Numbers Japheth Wood, PhD Fibonacci Book Rabbit Problem Some problems to solve Solutions Binet’s Formula Fibonacci’s](https://reader034.fdocuments.us/reader034/viewer/2022051410/60288b3a40297e1cab780ce5/html5/thumbnails/24.jpg)
FibonacciNumbers
Japheth Wood,PhD
Fibonacci BookRabbit Problem
Some problems tosolve
Solutions
Binet’s Formula
How many increasing paths are therethrough the honeycomb from 1 to 7?
· · ·
Path(5)+Path(6)=Path(7)Path(n − 2) + Path(n − 1) = Path(n)
n 1 2 3 4 5 6 7P(n) 1 1 2
3 5 8 13
Paths ending 5-7 Paths ending 6-7
![Page 25: Fibonacci Numbers - Bard Math Circle · 11/12/2019 · Fibonacci Numbers Japheth Wood, PhD Fibonacci Book Rabbit Problem Some problems to solve Solutions Binet’s Formula Fibonacci’s](https://reader034.fdocuments.us/reader034/viewer/2022051410/60288b3a40297e1cab780ce5/html5/thumbnails/25.jpg)
FibonacciNumbers
Japheth Wood,PhD
Fibonacci BookRabbit Problem
Some problems tosolve
Solutions
Binet’s Formula
How many increasing paths are therethrough the honeycomb from 1 to 7?
· · ·
Path(5)+Path(6)=Path(7)Path(n − 2) + Path(n − 1) = Path(n)
n 1 2 3 4 5 6 7P(n) 1 1 2 3
5 8 13
Paths ending 5-7 Paths ending 6-7
![Page 26: Fibonacci Numbers - Bard Math Circle · 11/12/2019 · Fibonacci Numbers Japheth Wood, PhD Fibonacci Book Rabbit Problem Some problems to solve Solutions Binet’s Formula Fibonacci’s](https://reader034.fdocuments.us/reader034/viewer/2022051410/60288b3a40297e1cab780ce5/html5/thumbnails/26.jpg)
FibonacciNumbers
Japheth Wood,PhD
Fibonacci BookRabbit Problem
Some problems tosolve
Solutions
Binet’s Formula
How many increasing paths are therethrough the honeycomb from 1 to 7?
· · ·
Path(5)+Path(6)=Path(7)Path(n − 2) + Path(n − 1) = Path(n)
n 1 2 3 4 5 6 7P(n) 1 1 2 3 5
8 13
Paths ending 5-7 Paths ending 6-7
![Page 27: Fibonacci Numbers - Bard Math Circle · 11/12/2019 · Fibonacci Numbers Japheth Wood, PhD Fibonacci Book Rabbit Problem Some problems to solve Solutions Binet’s Formula Fibonacci’s](https://reader034.fdocuments.us/reader034/viewer/2022051410/60288b3a40297e1cab780ce5/html5/thumbnails/27.jpg)
FibonacciNumbers
Japheth Wood,PhD
Fibonacci BookRabbit Problem
Some problems tosolve
Solutions
Binet’s Formula
How many increasing paths are therethrough the honeycomb from 1 to 7?
· · ·
Path(5)+Path(6)=Path(7)Path(n − 2) + Path(n − 1) = Path(n)
n 1 2 3 4 5 6 7P(n) 1 1 2 3 5 8
13
Paths ending 5-7 Paths ending 6-7
![Page 28: Fibonacci Numbers - Bard Math Circle · 11/12/2019 · Fibonacci Numbers Japheth Wood, PhD Fibonacci Book Rabbit Problem Some problems to solve Solutions Binet’s Formula Fibonacci’s](https://reader034.fdocuments.us/reader034/viewer/2022051410/60288b3a40297e1cab780ce5/html5/thumbnails/28.jpg)
FibonacciNumbers
Japheth Wood,PhD
Fibonacci BookRabbit Problem
Some problems tosolve
Solutions
Binet’s Formula
How many increasing paths are therethrough the honeycomb from 1 to 7?
· · ·
Path(5)+Path(6)=Path(7)Path(n − 2) + Path(n − 1) = Path(n)
n 1 2 3 4 5 6 7P(n) 1 1 2 3 5 8 13
Paths ending 5-7 Paths ending 6-7
![Page 29: Fibonacci Numbers - Bard Math Circle · 11/12/2019 · Fibonacci Numbers Japheth Wood, PhD Fibonacci Book Rabbit Problem Some problems to solve Solutions Binet’s Formula Fibonacci’s](https://reader034.fdocuments.us/reader034/viewer/2022051410/60288b3a40297e1cab780ce5/html5/thumbnails/29.jpg)
FibonacciNumbers
Japheth Wood,PhD
Fibonacci BookRabbit Problem
Some problems tosolve
Solutions
Binet’s Formula
The Fibonacci Sequence
F (n) = F (n − 1) + F (n − 2) (for n > 2), F (1) = 1, F (2) = 1
n 0 1 2 3 4 5 6 · · ·F (n) 0 1 1 2 3 5 8 · · ·
A Fibonacci-ish Sequence (Gibonacci?)
G (n) = G (n − 1) + G (n − 2)
n 0 1 2 3 4 5 6 · · ·G (n) 4 -2 2 0 2 2 4 · · ·
![Page 30: Fibonacci Numbers - Bard Math Circle · 11/12/2019 · Fibonacci Numbers Japheth Wood, PhD Fibonacci Book Rabbit Problem Some problems to solve Solutions Binet’s Formula Fibonacci’s](https://reader034.fdocuments.us/reader034/viewer/2022051410/60288b3a40297e1cab780ce5/html5/thumbnails/30.jpg)
FibonacciNumbers
Japheth Wood,PhD
Fibonacci BookRabbit Problem
Some problems tosolve
Solutions
Binet’s Formula
The Fibonacci Sequence
F (n) = F (n − 1) + F (n − 2) (for n > 2), F (1) = 1, F (2) = 1
n 0 1 2 3 4 5 6 · · ·F (n) 0 1 1 2 3 5 8 · · ·
A Fibonacci-ish Sequence (Gibonacci?)
G (n) = G (n − 1) + G (n − 2)
n 0 1 2 3 4 5 6 · · ·G (n) 4 -2 2 0 2 2 4 · · ·
![Page 31: Fibonacci Numbers - Bard Math Circle · 11/12/2019 · Fibonacci Numbers Japheth Wood, PhD Fibonacci Book Rabbit Problem Some problems to solve Solutions Binet’s Formula Fibonacci’s](https://reader034.fdocuments.us/reader034/viewer/2022051410/60288b3a40297e1cab780ce5/html5/thumbnails/31.jpg)
FibonacciNumbers
Japheth Wood,PhD
Fibonacci BookRabbit Problem
Some problems tosolve
Solutions
Binet’s Formula
Fact 1: Scaling a Fibonacci-ish Sequence yields a · · ·
n 0 1 2 3 4 5 6 · · ·G (n) 4 -3 1 -2 -1 -3 -4 · · ·4G (n) 16 -12 4 -8 -4 -12 -16 · · ·
Fact 2: If G (0) is 0 then . . .
n 0 1 2 3 4 5 6 · · ·G (n) 0 3 3 6 9 15 24 · · ·
F (n) 0 1 1 2 3 5 8 · · ·
Fact 3: Subtracting Fibonacci-ish Sequences yields a . . .
n 0 1 2 3 4 5 · · ·G (n) 4 3 7 10 17 27 · · ·H(n) 2 1 3 4 7 11 · · ·
G (n)− H(n) 2 2 4 6 10 16 · · ·
![Page 32: Fibonacci Numbers - Bard Math Circle · 11/12/2019 · Fibonacci Numbers Japheth Wood, PhD Fibonacci Book Rabbit Problem Some problems to solve Solutions Binet’s Formula Fibonacci’s](https://reader034.fdocuments.us/reader034/viewer/2022051410/60288b3a40297e1cab780ce5/html5/thumbnails/32.jpg)
FibonacciNumbers
Japheth Wood,PhD
Fibonacci BookRabbit Problem
Some problems tosolve
Solutions
Binet’s Formula
Fact 1: Scaling a Fibonacci-ish Sequence yields a · · ·
n 0 1 2 3 4 5 6 · · ·G (n) 4 -3 1 -2 -1 -3 -4 · · ·4G (n) 16 -12 4 -8 -4 -12 -16 · · ·
Fact 2: If G (0) is 0 then . . .
n 0 1 2 3 4 5 6 · · ·G (n) 0 3 3 6 9 15 24 · · ·
F (n) 0 1 1 2 3 5 8 · · ·
Fact 3: Subtracting Fibonacci-ish Sequences yields a . . .
n 0 1 2 3 4 5 · · ·G (n) 4 3 7 10 17 27 · · ·H(n) 2 1 3 4 7 11 · · ·
G (n)− H(n) 2 2 4 6 10 16 · · ·
![Page 33: Fibonacci Numbers - Bard Math Circle · 11/12/2019 · Fibonacci Numbers Japheth Wood, PhD Fibonacci Book Rabbit Problem Some problems to solve Solutions Binet’s Formula Fibonacci’s](https://reader034.fdocuments.us/reader034/viewer/2022051410/60288b3a40297e1cab780ce5/html5/thumbnails/33.jpg)
FibonacciNumbers
Japheth Wood,PhD
Fibonacci BookRabbit Problem
Some problems tosolve
Solutions
Binet’s Formula
Fact 1: Scaling a Fibonacci-ish Sequence yields a · · ·
n 0 1 2 3 4 5 6 · · ·G (n) 4 -3 1 -2 -1 -3 -4 · · ·4G (n) 16 -12 4 -8 -4 -12 -16 · · ·
Fact 2: If G (0) is 0 then . . .
n 0 1 2 3 4 5 6 · · ·G (n) 0 3 3 6 9 15 24 · · ·F (n) 0 1 1 2 3 5 8 · · ·
Fact 3: Subtracting Fibonacci-ish Sequences yields a . . .
n 0 1 2 3 4 5 · · ·G (n) 4 3 7 10 17 27 · · ·H(n) 2 1 3 4 7 11 · · ·
G (n)− H(n) 2 2 4 6 10 16 · · ·
![Page 34: Fibonacci Numbers - Bard Math Circle · 11/12/2019 · Fibonacci Numbers Japheth Wood, PhD Fibonacci Book Rabbit Problem Some problems to solve Solutions Binet’s Formula Fibonacci’s](https://reader034.fdocuments.us/reader034/viewer/2022051410/60288b3a40297e1cab780ce5/html5/thumbnails/34.jpg)
FibonacciNumbers
Japheth Wood,PhD
Fibonacci BookRabbit Problem
Some problems tosolve
Solutions
Binet’s Formula
Fact 1: Scaling a Fibonacci-ish Sequence yields a · · ·
n 0 1 2 3 4 5 6 · · ·G (n) 4 -3 1 -2 -1 -3 -4 · · ·4G (n) 16 -12 4 -8 -4 -12 -16 · · ·
Fact 2: If G (0) is 0 then . . .
n 0 1 2 3 4 5 6 · · ·G (n) 0 3 3 6 9 15 24 · · ·F (n) 0 1 1 2 3 5 8 · · ·
Fact 3: Subtracting Fibonacci-ish Sequences yields a . . .
n 0 1 2 3 4 5 · · ·G (n) 4 3 7 10 17 27 · · ·H(n) 2 1 3 4 7 11 · · ·
G (n)− H(n) 2 2 4 6 10 16 · · ·
![Page 35: Fibonacci Numbers - Bard Math Circle · 11/12/2019 · Fibonacci Numbers Japheth Wood, PhD Fibonacci Book Rabbit Problem Some problems to solve Solutions Binet’s Formula Fibonacci’s](https://reader034.fdocuments.us/reader034/viewer/2022051410/60288b3a40297e1cab780ce5/html5/thumbnails/35.jpg)
FibonacciNumbers
Japheth Wood,PhD
Fibonacci BookRabbit Problem
Some problems tosolve
Solutions
Binet’s Formula
Fact 1: Scaling a Fibonacci-ish Sequence yields a · · ·
n 0 1 2 3 4 5 6 · · ·G (n) 4 -3 1 -2 -1 -3 -4 · · ·4G (n) 16 -12 4 -8 -4 -12 -16 · · ·
Fact 2: If G (0) is 0 then . . .
n 0 1 2 3 4 5 6 · · ·G (n) 0 3 3 6 9 15 24 · · ·F (n) 0 1 1 2 3 5 8 · · ·
Fact 3: Subtracting Fibonacci-ish Sequences yields a . . .
n 0 1 2 3 4 5 · · ·G (n) 4 3 7 10 17 27 · · ·H(n) 2 1 3 4 7 11 · · ·
G (n)− H(n) 2 2 4 6 10 16 · · ·
![Page 36: Fibonacci Numbers - Bard Math Circle · 11/12/2019 · Fibonacci Numbers Japheth Wood, PhD Fibonacci Book Rabbit Problem Some problems to solve Solutions Binet’s Formula Fibonacci’s](https://reader034.fdocuments.us/reader034/viewer/2022051410/60288b3a40297e1cab780ce5/html5/thumbnails/36.jpg)
FibonacciNumbers
Japheth Wood,PhD
Fibonacci BookRabbit Problem
Some problems tosolve
Solutions
Binet’s Formula
Two interesting Fibonacci-ish sequences
n 0 1 2 3 4 · · ·R(n) 1 r r2 r3 r4 · · ·S(n) 1 s s2 s3 s4 · · ·
R(n)− S(n) 0 r−s r2−s2 r3−s3 r4−s4 · · ·
F (n) 0 1 r2−s2r−s
r3−s3r−s
r4−s4r−s · · ·
Necessary (and Sufficient) Conditions
R(0) + R(1) = R(2) or 1 + r = r2 and also 1 + s = s2
Both r and s are solutions of 1 + x = x2.r = 1+
√5
2 ≈ 1.618 and s = 1−√5
2 ≈ −0.618
Binet’s Formula
F (n) =rn − sn
r − s=
(1+√5
2
)n−
(1−√5
2
)n
√5
![Page 37: Fibonacci Numbers - Bard Math Circle · 11/12/2019 · Fibonacci Numbers Japheth Wood, PhD Fibonacci Book Rabbit Problem Some problems to solve Solutions Binet’s Formula Fibonacci’s](https://reader034.fdocuments.us/reader034/viewer/2022051410/60288b3a40297e1cab780ce5/html5/thumbnails/37.jpg)
FibonacciNumbers
Japheth Wood,PhD
Fibonacci BookRabbit Problem
Some problems tosolve
Solutions
Binet’s Formula
Two interesting Fibonacci-ish sequences
n 0 1 2 3 4 · · ·R(n) 1 r r2 r3 r4 · · ·S(n) 1 s s2 s3 s4 · · ·
R(n)− S(n) 0 r−s r2−s2 r3−s3 r4−s4 · · ·
F (n) 0 1 r2−s2r−s
r3−s3r−s
r4−s4r−s · · ·
Necessary (and Sufficient) Conditions
R(0) + R(1) = R(2) or 1 + r = r2 and also 1 + s = s2
Both r and s are solutions of 1 + x = x2.r = 1+
√5
2 ≈ 1.618 and s = 1−√5
2 ≈ −0.618
Binet’s Formula
F (n) =rn − sn
r − s=
(1+√5
2
)n−
(1−√5
2
)n
√5
![Page 38: Fibonacci Numbers - Bard Math Circle · 11/12/2019 · Fibonacci Numbers Japheth Wood, PhD Fibonacci Book Rabbit Problem Some problems to solve Solutions Binet’s Formula Fibonacci’s](https://reader034.fdocuments.us/reader034/viewer/2022051410/60288b3a40297e1cab780ce5/html5/thumbnails/38.jpg)
FibonacciNumbers
Japheth Wood,PhD
Fibonacci BookRabbit Problem
Some problems tosolve
Solutions
Binet’s Formula
Two interesting Fibonacci-ish sequences
n 0 1 2 3 4 · · ·R(n) 1 r r2 r3 r4 · · ·S(n) 1 s s2 s3 s4 · · ·
R(n)− S(n) 0 r−s r2−s2 r3−s3 r4−s4 · · ·
F (n) 0 1 r2−s2r−s
r3−s3r−s
r4−s4r−s · · ·
Necessary (and Sufficient) Conditions
R(0) + R(1) = R(2) or 1 + r = r2 and also 1 + s = s2
Both r and s are solutions of 1 + x = x2.r = 1+
√5
2 ≈ 1.618 and s = 1−√5
2 ≈ −0.618
Binet’s Formula
F (n) =rn − sn
r − s=
(1+√5
2
)n−
(1−√5
2
)n
√5
![Page 39: Fibonacci Numbers - Bard Math Circle · 11/12/2019 · Fibonacci Numbers Japheth Wood, PhD Fibonacci Book Rabbit Problem Some problems to solve Solutions Binet’s Formula Fibonacci’s](https://reader034.fdocuments.us/reader034/viewer/2022051410/60288b3a40297e1cab780ce5/html5/thumbnails/39.jpg)
FibonacciNumbers
Japheth Wood,PhD
Fibonacci BookRabbit Problem
Some problems tosolve
Solutions
Binet’s Formula
Two interesting Fibonacci-ish sequences
n 0 1 2 3 4 · · ·R(n) 1 r r2 r3 r4 · · ·S(n) 1 s s2 s3 s4 · · ·
R(n)− S(n) 0 r−s r2−s2 r3−s3 r4−s4 · · ·
F (n) 0 1 r2−s2r−s
r3−s3r−s
r4−s4r−s · · ·
Necessary (and Sufficient) Conditions
R(0) + R(1) = R(2) or 1 + r = r2 and also 1 + s = s2
Both r and s are solutions of 1 + x = x2.
r = 1+√5
2 ≈ 1.618 and s = 1−√5
2 ≈ −0.618
Binet’s Formula
F (n) =rn − sn
r − s=
(1+√5
2
)n−
(1−√5
2
)n
√5
![Page 40: Fibonacci Numbers - Bard Math Circle · 11/12/2019 · Fibonacci Numbers Japheth Wood, PhD Fibonacci Book Rabbit Problem Some problems to solve Solutions Binet’s Formula Fibonacci’s](https://reader034.fdocuments.us/reader034/viewer/2022051410/60288b3a40297e1cab780ce5/html5/thumbnails/40.jpg)
FibonacciNumbers
Japheth Wood,PhD
Fibonacci BookRabbit Problem
Some problems tosolve
Solutions
Binet’s Formula
Two interesting Fibonacci-ish sequences
n 0 1 2 3 4 · · ·R(n) 1 r r2 r3 r4 · · ·S(n) 1 s s2 s3 s4 · · ·
R(n)− S(n) 0 r−s r2−s2 r3−s3 r4−s4 · · ·
F (n) 0 1 r2−s2r−s
r3−s3r−s
r4−s4r−s · · ·
Necessary (and Sufficient) Conditions
R(0) + R(1) = R(2) or 1 + r = r2 and also 1 + s = s2
Both r and s are solutions of 1 + x = x2.r = 1+
√5
2 ≈ 1.618 and s = 1−√5
2 ≈ −0.618
Binet’s Formula
F (n) =rn − sn
r − s=
(1+√5
2
)n−
(1−√5
2
)n
√5
![Page 41: Fibonacci Numbers - Bard Math Circle · 11/12/2019 · Fibonacci Numbers Japheth Wood, PhD Fibonacci Book Rabbit Problem Some problems to solve Solutions Binet’s Formula Fibonacci’s](https://reader034.fdocuments.us/reader034/viewer/2022051410/60288b3a40297e1cab780ce5/html5/thumbnails/41.jpg)
FibonacciNumbers
Japheth Wood,PhD
Fibonacci BookRabbit Problem
Some problems tosolve
Solutions
Binet’s Formula
Two interesting Fibonacci-ish sequences
n 0 1 2 3 4 · · ·R(n) 1 r r2 r3 r4 · · ·S(n) 1 s s2 s3 s4 · · ·
R(n)− S(n) 0 r−s r2−s2 r3−s3 r4−s4 · · ·
F (n) 0 1 r2−s2r−s
r3−s3r−s
r4−s4r−s · · ·
Necessary (and Sufficient) Conditions
R(0) + R(1) = R(2) or 1 + r = r2 and also 1 + s = s2
Both r and s are solutions of 1 + x = x2.r = 1+
√5
2 ≈ 1.618 and s = 1−√5
2 ≈ −0.618
Binet’s Formula
F (n) =rn − sn
r − s=
(1+√5
2
)n−
(1−√5
2
)n
√5
![Page 42: Fibonacci Numbers - Bard Math Circle · 11/12/2019 · Fibonacci Numbers Japheth Wood, PhD Fibonacci Book Rabbit Problem Some problems to solve Solutions Binet’s Formula Fibonacci’s](https://reader034.fdocuments.us/reader034/viewer/2022051410/60288b3a40297e1cab780ce5/html5/thumbnails/42.jpg)
FibonacciNumbers
Japheth Wood,PhD
Fibonacci BookRabbit Problem
Some problems tosolve
Solutions
Binet’s Formula
Two interesting Fibonacci-ish sequences
n 0 1 2 3 4 · · ·R(n) 1 r r2 r3 r4 · · ·S(n) 1 s s2 s3 s4 · · ·
R(n)− S(n) 0 r−s r2−s2 r3−s3 r4−s4 · · ·F (n) 0 1 r2−s2
r−sr3−s3r−s
r4−s4r−s · · ·
Necessary (and Sufficient) Conditions
R(0) + R(1) = R(2) or 1 + r = r2 and also 1 + s = s2
Both r and s are solutions of 1 + x = x2.r = 1+
√5
2 ≈ 1.618 and s = 1−√5
2 ≈ −0.618
Binet’s Formula
F (n) =rn − sn
r − s=
(1+√5
2
)n−
(1−√5
2
)n
√5
![Page 43: Fibonacci Numbers - Bard Math Circle · 11/12/2019 · Fibonacci Numbers Japheth Wood, PhD Fibonacci Book Rabbit Problem Some problems to solve Solutions Binet’s Formula Fibonacci’s](https://reader034.fdocuments.us/reader034/viewer/2022051410/60288b3a40297e1cab780ce5/html5/thumbnails/43.jpg)
FibonacciNumbers
Japheth Wood,PhD
Fibonacci BookRabbit Problem
Some problems tosolve
Solutions
Binet’s Formula
Two interesting Fibonacci-ish sequences
n 0 1 2 3 4 · · ·R(n) 1 r r2 r3 r4 · · ·S(n) 1 s s2 s3 s4 · · ·
R(n)− S(n) 0 r−s r2−s2 r3−s3 r4−s4 · · ·F (n) 0 1 r2−s2
r−sr3−s3r−s
r4−s4r−s · · ·
Necessary (and Sufficient) Conditions
R(0) + R(1) = R(2) or 1 + r = r2 and also 1 + s = s2
Both r and s are solutions of 1 + x = x2.r = 1+
√5
2 ≈ 1.618 and s = 1−√5
2 ≈ −0.618
Binet’s Formula
F (n) =rn − sn
r − s=
(1+√5
2
)n−(1−√5
2
)n
√5
![Page 44: Fibonacci Numbers - Bard Math Circle · 11/12/2019 · Fibonacci Numbers Japheth Wood, PhD Fibonacci Book Rabbit Problem Some problems to solve Solutions Binet’s Formula Fibonacci’s](https://reader034.fdocuments.us/reader034/viewer/2022051410/60288b3a40297e1cab780ce5/html5/thumbnails/44.jpg)
FibonacciNumbers
Japheth Wood,PhD
Fibonacci BookRabbit Problem
Some problems tosolve
Solutions
Binet’s Formula
The End
Thank You!Japheth Wood 〈[email protected]〉