![Page 1: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/1.jpg)
Fibonacci Numbers, Polynomial Coefficients, and
Vector Programs.
![Page 2: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/2.jpg)
Leonardo Fibonacci
In 1202, Fibonacci proposed a problem about the growth of rabbit populations.
![Page 3: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/3.jpg)
Inductive Definition or Recurrence Relation for the
Fibonacci Numbers Stage 0, Initial Condition, or Base Case:Fib(0) = 0; Fib (1) = 1
Inductive RuleFor n>1, Fib(n) = Fib(n-1) + Fib(n-2)
n 0 1 2 3 4 5 6 7
Fib(n) 0 1 1 2 3 5 813
![Page 4: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/4.jpg)
Sneezwort (Achilleaptarmica)
Each time the plant starts a new shoot it takes two months before it is strong
enough to support branching.
![Page 5: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/5.jpg)
Counting Petals
5 petals: buttercup, wild rose, larkspur,columbine (aquilegia)
8 petals: delphiniums 13 petals: ragwort, corn marigold, cineraria,
some daisies 21 petals: aster, black-eyed susan, chicory 34 petals: plantain, pyrethrum 55, 89 petals: michaelmas daisies, the
asteraceae family.
![Page 6: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/6.jpg)
Pineapple whorls
Church and Turing were both interested in the number of whorls in each ring of the spiral.
The ratio of consecutive ring lengths approaches the Golden Ratio.
![Page 7: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/7.jpg)
![Page 8: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/8.jpg)
![Page 9: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/9.jpg)
AC
AB
AB
BCAC
BCAC
BC
AB
BC
BC
BC1
1 0
2
2
2
Definition of (Euclid)
Ratio obtained when you divide a line segment into two unequal parts such that the ratio of the whole to the larger part is the same as the ratio of the larger to the smaller.
A B C
![Page 10: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/10.jpg)
Expanding Recursively
11
11
11
11
11
11
![Page 11: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/11.jpg)
Continued Fraction Representation
11
11
11
11
11
11
11
11
11
1 ....
![Page 12: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/12.jpg)
Continued Fraction Representation
11
11
11
11
11
11
11
11
11
1 ....
1 52
![Page 13: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/13.jpg)
1,1,2,3,5,8,13,21,34,55,….
2/1 = 23/2 = 1.55/3 = 1.666…8/5 = 1.613/8 = 1.62521/13 = 1.6153846…34/21 = 1.61904…
= 1.6180339887498948482045
![Page 14: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/14.jpg)
How to divide polynomials?
1 1
1 – X?
1 – X 1
1
-(1 – X)
X
+ X
-(X – X2)
X2
+ X2
-(X2 – X3)
X3
=1 + X + X2 + X3 + X4 + X5 + X6 + X7 + …
…
![Page 15: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/15.jpg)
1 + X1 + X11 + X + X22 + X + X33 + … + X + … + Xnn + ….. = + ….. = 11
1 - X1 - X
The Infinite Geometric Series
![Page 16: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/16.jpg)
(1-X) ( 1 + X1 + X2 + X 3 + … + Xn + … )= 1 + X1 + X2 + X 3 + … + Xn + Xn+1
+ …. - X1 - X2 - X 3 - … - Xn-1 – Xn - Xn+1 - …
= 1
1 + X1 + X11 + X + X22 + X + X33 + … + X + … + Xnn + ….. = + ….. = 11
1 - X1 - X
![Page 17: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/17.jpg)
1 + X1 + X11 + X + X22 + X + X33 + … + X + … + Xnn + ….. = + ….. = 11
1 - X1 - X
1 – X 1
1
-(1 – X)
X
+ X
-(X – X2)
X2
+ X2 + …
-(X2 – X3)
X3…
![Page 18: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/18.jpg)
X X 1 – X – X2
Something a bit more complicated
1 – X – X2 X
X2 + X3
-(X – X2 – X3)
X
2X3 + X4
-(X2 – X3 – X4)
+ X2
-(2X3 – 2X4 – 2X5)
+ 2X3
3X4 + 2X5
+ 3X4
-(3X4 – 3X5 – 3X6)
5X5 + 3X6
+ 5X5
-(5X5 – 5X6 – 5X7)
8X6 + 5X7
+ 8X6
-(8X6 – 8X7 – 8X8)
![Page 19: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/19.jpg)
Hence
= F0 1 + F1 X1 + F2 X2 +F3 X3 + F4 X4 +
F5 X5 + F6 X6 + …
X X
1 – X – X2
= 01 + 1 X1 + 1 X2 + 2X3 + 3X4 + 5X5 + 8X6 + …
![Page 20: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/20.jpg)
Going the Other Way
(1 - X- X2) ( F0 1 + F1 X1 + F2 X2 + … + Fn-2 Xn-2 + Fn-1 Xn-1 + Fn Xn + …
F0 = 0, F1 = 1
![Page 21: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/21.jpg)
Going the Other Way
(1 - X- X2) ( F0 1 + F1 X1 + F2 X2 + … + Fn-2 Xn-2 + Fn-1 Xn-1 + Fn Xn + …
= ( F0 1 + F1 X1 + F2 X2 + … + Fn-2 Xn-2 + Fn-1 Xn-1 + Fn Xn + …
- F0 X1 - F1 X2 - … - Fn-3 Xn-2 - Fn-2 Xn-1 - Fn-1 Xn - … - F0 X2 - … - Fn-4 Xn-2 - Fn-3 Xn-1 - Fn-2 Xn - …
= F0 1 + ( F1 – F0 ) X1 F0 = 0, F1 = 1= X
![Page 22: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/22.jpg)
Thus
F0 1 + F1 X1 + F2 X2 + … + Fn-1 Xn-1 + Fn Xn + …
X X
1 – X – X2=
= X/(1- X)(1 – (-)-1 X)
-1 = 1, --1=1
![Page 23: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/23.jpg)
XX
(1 – (1 – X)(1- (-X)(1- (-))--
11X)X)
Linear factors on the bottom
n=0..∞= Xn?
![Page 24: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/24.jpg)
(1 + aX1 + a2X2 + … + anXn + …..) (1 + bX1 + b2X2 + … + bnXn
+ …..) =
11
(1 – aX)(1-bX)(1 – aX)(1-bX)
Geometric Series (Quadratic Form)
aan+1 n+1 – – bbn+1n+1
aa - b- b
n=0..∞ Xn
=
=
![Page 25: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/25.jpg)
11
(1 – (1 – X)(1- (-X)(1- (---
11X))X))
Geometric Series (Quadratic Form)
n+1 n+1 – (-– (--1-1))n+1n+1
√√5 5 n=0.. ∞
= Xn
![Page 26: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/26.jpg)
XX
(1 – (1 – X)(1- (-X)(1- (--1-1X)X)
Power Series Expansion of F
n+1 n+1 – (-– (---1)1)n+1n+1
√√55 n=0.. ∞
= Xn+1
![Page 27: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/27.jpg)
0 1 2 30 1 2 32
01i
ii
xF x F x F x F x F x
x x
20
1 1
1 5
i
i i
i
xx
x x
![Page 28: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/28.jpg)
The ith Fibonacci number is:
0
1 1
5
i
i
i
Leonhard Euler (1765) J. P. M. Binet (1843)A de Moivre (1730)
![Page 29: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/29.jpg)
(1 + aX1 + a2X2 + … + anXn + …..) (1 + bX1 + b2X2 + … + bnXn
+ …..) =
11
(1 – aX)(1-bX)(1 – aX)(1-bX)
Let’s Derive This
aan+1 n+1 – – bbn+1n+1
aa - b- b
n=0..∞ Xn
=
=
![Page 30: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/30.jpg)
1 + Y1 + Y11 + Y + Y22 + Y + Y 33 + … + Y + … + Ynn + ….. = + ….. = 11
1 - Y1 - Y
Substituting Y = aX …
![Page 31: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/31.jpg)
1 + aX1 + aX11 + a + a22XX22 + a + a33X X 33 + … + a + … + annXXnn + ….. =+ ….. =
11
1 - aX1 - aX
Geometric Series (Linear Form)
![Page 32: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/32.jpg)
(1 + aX1 + a2X2 + … + anXn + …..) (1 + bX1 + b2X2 + … + bnXn
+ …..) =
11
(1 – aX)(1-bX)(1 – aX)(1-bX)
Geometric Series (Quadratic Form)
![Page 33: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/33.jpg)
(1 + aX1 + a2X2 + … + anXn + …..) (1 + bX1 + b2X2 + … + bnXn
+ …..) =
1 + c1X1 + .. + ck Xk + …
Suppose we multiply this out to get a single, infinite
polynomial.
What is an expression for Cn?
![Page 34: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/34.jpg)
(1 + aX1 + a2X2 + … + anXn + …..) (1 + bX1 + b2X2 + … + bnXn
+ …..) =
1 + c1X1 + .. + ck Xk + …
cn =
aa00bbnn + a + a11bbn-1n-1 +… a +… aiibbn-in-i… + a… + an-1n-1bb11 + +
aannbb00
![Page 35: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/35.jpg)
(1 + aX1 + a2X2 + … + anXn + …..) (1 + bX1 + b2X2 + … + bnXn
+ …..) =
1 + c1X1 + .. + ck Xk + …
If a = b then
cn = (n+1)(an)
aa00bbnn + a + a11bbn-1n-1 +… a +… aiibbn-in-i… + a… + an-1n-1bb11 + +
aannbb00
![Page 36: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/36.jpg)
(a-b) (aa00bbnn + a + a11bbn-1n-1 +… a +… aiibbn-in-i… + a… + an-1n-1bb11 + a + annbb00)
= aa11bbnn +… a +… ai+1i+1bbn-in-i… + a… + annbb11 + a + an+1n+1bb00
- aa00bbn+1 n+1 – a – a11bbnn… a… ai+1i+1bbn-in-i… - a… - an-1n-1bb22 - a - annbb11
= - bn+1 + an+1
= an+1 – bn+1
aa00bbnn + a + a11bbn-1n-1 +… a +… aiibbn-in-i… + a… + an-1n-1bb11 + a + annbb00 = = aan+1 n+1 – – bbn+1n+1
aa - b- b
![Page 37: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/37.jpg)
(1 + aX1 + a2X2 + … + anXn + …..) (1 + bX1 + b2X2 + … + bnXn
+ …..) =
1 + c1X1 + .. + ck Xk + …
if a b then
cn =
aa00bbnn + a + a11bbn-1n-1 +… a +… aiibbn-in-i… + a… + an-1n-1bb11 + +
aannbb00
aan+1 n+1 – – bbn+1n+1
aa - b- b
![Page 38: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/38.jpg)
(1 + aX1 + a2X2 + … + anXn + …..) (1 + bX1 + b2X2 + … + bnXn
+ …..) =
11
(1 – aX)(1-bX)(1 – aX)(1-bX)
Geometric Series (Quadratic Form)
aan+1 n+1 – – bbn+1n+1
aa - b- b
n=0..Xn
=
=
n=0.. Xnor (n+1)a(n+1)ann
when a=b
![Page 39: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/39.jpg)
Sequences That Sum To n
Let fn+1 be the number of different sequences of 1’s and 2’s that sum to n.
Example: f5 = 5
![Page 40: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/40.jpg)
Sequences That Sum To n
Let fn+1 be the number of different sequences of 1’s and 2’s that sum to n.
Example: f5 = 5
4 = 2 + 2
2 + 1 + 11 + 2 + 11 + 1 + 21 + 1 + 1 + 1
![Page 41: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/41.jpg)
Sequences That Sum To n
f1
f2
f3
Let fn+1 be the number of different sequences of 1’s and 2’s that sum to n.
![Page 42: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/42.jpg)
Sequences That Sum To n
f1 = 1
0 = the empty sumf2 = 1
1 = 1
f3 = 2
2 = 1 + 1
2
Let fn+1 be the number of different sequences of 1’s and 2’s that sum to n.
![Page 43: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/43.jpg)
Sequences That Sum To n
fn+1 = fn + fn-1
Let fn+1 be the number of different sequences of 1’s and 2’s that sum to n.
![Page 44: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/44.jpg)
Sequences That Sum To n
fn+1 = fn + fn-1
Let fn+1 be the number of different sequences of 1’s and 2’s that sum to n.
# of sequences beginning with a 2
# of sequences beginning with a 1
![Page 45: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/45.jpg)
Fibonacci Numbers Again
ffn+1n+1 = f = fnn + f + fn-1n-1
ff11 = 1 f = 1 f22 = 1 = 1
Let fn+1 be the number of different sequences of 1’s and 2’s that sum to
n.
![Page 46: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/46.jpg)
Visual Representation: Tiling
Let fn+1 be the number of different ways to tile a 1 × n strip with squares and dominoes.
![Page 47: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/47.jpg)
Visual Representation: Tiling
Let fn+1 be the number of different ways to tile a 1 × n strip with squares and dominoes.
![Page 48: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/48.jpg)
Visual Representation: Tiling
1 way to tile a strip of length 0
1 way to tile a strip of length 1:
2 ways to tile a strip of length 2:
![Page 49: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/49.jpg)
fn+1 = fn + fn-1
fn+1 is number of ways to tile length n.
fn tilings that start with a square.
fn-1 tilings that start with a domino.
![Page 50: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/50.jpg)
Let’s use this visual representation to prove a couple of
Fibonacci identities.
![Page 51: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/51.jpg)
Fibonacci Identities
Some examples:
F2n = F1 + F3 + F5 + … + F2n-1
Fm+n+1 = Fm+1 Fn+1 + Fm Fn
(Fn)2 = Fn-1 Fn+1 + (-1)n
![Page 52: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/52.jpg)
Fm+n+1 = Fm+1 Fn+1 + Fm Fn
mm nn
m-1m-1 n-1n-1
![Page 53: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/53.jpg)
(Fn)2 = Fn-1 Fn+1 + (-1)n
![Page 54: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/54.jpg)
(Fn)2 = Fn-1 Fn+1 + (-1)n
n-1n-1
Fn tilings of a strip of length n-1
![Page 55: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/55.jpg)
(Fn)2 = Fn-1 Fn+1 + (-1)n
n-1n-1
n-1n-1
![Page 56: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/56.jpg)
(Fn)2 = Fn-1 Fn+1 + (-1)n
nn
(Fn)2 tilings of two strips of size n-1
![Page 57: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/57.jpg)
(Fn)2 = Fn-1 Fn+1 + (-1)n
nn
Draw a vertical “fault Draw a vertical “fault line” at the line” at the rightmost rightmost
position position (<n)(<n) possible without possible without
cutting any cutting any dominoes dominoes
![Page 58: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/58.jpg)
(Fn)2 = Fn-1 Fn+1 + (-1)n
nn
Swap the tailsSwap the tails at the at the fault linefault line to map to a to map to a tiling of 2 n-1 ‘s to a tiling of 2 n-1 ‘s to a tiling of an n-2 and an tiling of an n-2 and an n.n.
![Page 59: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/59.jpg)
(Fn)2 = Fn-1 Fn+1 + (-1)n
nn
Swap the tailsSwap the tails at the at the fault linefault line to map to a to map to a tiling of 2 n-1 ‘s to a tiling of 2 n-1 ‘s to a tiling of an n-2 and an tiling of an n-2 and an n.n.
![Page 60: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/60.jpg)
(Fn)2 = Fn-1 Fn+1 + (-1)n-1
n n eveneven
n oddn odd
![Page 61: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/61.jpg)
The Fibonacci Quarterly
![Page 62: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/62.jpg)
Vector Programs
Let’s define a (parallel) programming language called VECTOR that operates on possibly infinite vectors of numbers. Each variable V! can be thought of as:
< * , * , * , * , *, *, . . . . . . . . . >
0 1 2 3 4 5 . . . . . . . . .
![Page 63: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/63.jpg)
Vector Programs
Let k stand for a scalar constant<k> will stand for the vector <k,0,0,0,…>
<0> = <0,0,0,0,….><1> = <1,0,0,0,…>
V! + T! means to add the vectors position-wise.
<4,2,3,…> + <5,1,1,….> = <9,3,4,…>
![Page 64: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/64.jpg)
Vector Programs
RIGHT(V!) means to shift every number in V! one position to the right and to place a 0 in position 0.
RIGHT( <1,2,3, …> ) = <0,1,2,3,. …>
![Page 65: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/65.jpg)
Vector Programs
Example:
V! := <6>;V! := RIGHT(V!) + <42>;V! := RIGHT(V!) + <2>;V! := RIGHT(V!) + <13>;
VV!! = < 13, 2, 42, 6, 0, 0, 0, . . . > = < 13, 2, 42, 6, 0, 0, 0, . . . >
Store
V! = <6,0,0,0,..>V! = <42,6,0,0,..>V! = <2,42,6,0,..> V!
= <13,2,42,6,.>
![Page 66: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/66.jpg)
Vector Programs
Example:
V! := <1>;
Loop n times: V! := V! + RIGHT(V!);
VV!! = n = nthth row of Pascal’s triangle. row of Pascal’s triangle.
Store
V! = <1,0,0,0,..>
V! = <1,1,0,0,..>V! = <1,2,1,0,..> V! = <1,3,3,1,.>
![Page 67: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/67.jpg)
X1 X2 + + X3
Vector programs can be
implemented by polynomials!
![Page 68: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/68.jpg)
Programs -----> Polynomials
The vector V! = < a0, a1, a2, . . . > will be represented by the polynomial:
![Page 69: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/69.jpg)
Formal Power Series
The vector V! = < a0, a1, a2, . . . > will be represented by the formal power series:
![Page 70: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/70.jpg)
V! = < a0, a1, a2, . . . >
<0> is represented by 0<k> is represented by k
RIGHT(V! ) is represented by (PV X)
V! + T! is represented by (PV + PT)
![Page 71: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/71.jpg)
Vector Programs
Example:
V! := <1>;
Loop n times: V! := V! + RIGHT(V!);
VV!! = n = nthth row of Pascal’s triangle. row of Pascal’s triangle.
PV := 1;
PV := PV + PV X;
![Page 72: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/72.jpg)
Vector Programs
Example:
V! := <1>;
Loop n times: V! := V! + RIGHT(V!);
VV!! = n = nthth row of Pascal’s triangle. row of Pascal’s triangle.
PV := 1;
PV := PV (1+ X);
![Page 73: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/73.jpg)
Vector Programs
Example:
V! := <1>;
Loop n times: V! := V! + RIGHT(V!);
VV!! = n = nthth row of Pascal’s triangle. row of Pascal’s triangle.
PV = (1+ X)n
![Page 74: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/74.jpg)
Let’s add an instruction called PREFIXSUM to our
VECTOR language.
W! := PREFIXSUM(V!)
means that the ith position of W contains the sum of all the numbers in V from
positions 0 to i.
![Page 75: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/75.jpg)
What does this program output?
V! := 1! ;Loop k times: V! := PREFIXSUM(V!) ;
k’th Avenue01
23
4
![Page 76: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/76.jpg)
Al Karaji Perfect SquaresAl Karaji Perfect Squares
Zero_Ave := PREFIXSUM(<1>);First_Ave := PREFIXSUM(Zero_Ave);Second_Ave :=PREFIXSUM(First_Ave);
Output:= RIGHT(Second_Ave) + Second_Ave
Second_Ave = <1, 3, 6, 10, 15,. Second_Ave = <1, 3, 6, 10, 15,.
RIGHT(Second_Ave) = <0, 1, 3, 6, 10,.RIGHT(Second_Ave) = <0, 1, 3, 6, 10,.Output = <1, 4, 9, 16, 25
![Page 77: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/77.jpg)
Can you see how PREFIXSUM can be represented by a
familiar polynomial expression?
![Page 78: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/78.jpg)
W! := PREFIXSUM(V!)
is represented by
PW = PV / (1-X)
= PV (1+X+X2+X3+
….. )
![Page 79: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/79.jpg)
Al-Karaji Program
Zero_Ave = 1/(1-X);First_Ave = 1/(1-X)2;Second_Ave = 1/(1-X)3;
Output = 1/(1-X)2 + 2X/(1-X)3
(1-X)/(1-X)(1-X)/(1-X)3 3 + 2X/(1-X)+ 2X/(1-X)3 3
= (1+X)/(1-X)= (1+X)/(1-X)33
![Page 80: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/80.jpg)
(1+X)/(1-X)(1+X)/(1-X)33
Zero_Ave := PREFIXSUM(<1>);First_Ave := PREFIXSUM(Zero_Ave);Second_Ave :=PREFIXSUM(First_Ave);
Output:= RIGHT(Second_Ave) + Second_Ave
Second_Ave = <1, 3, 6, 10, 15,. Second_Ave = <1, 3, 6, 10, 15,.
RIGHT(Second_Ave) = <0, 1, 3, 6, 10,.RIGHT(Second_Ave) = <0, 1, 3, 6, 10,.Output = <1, 4, 9, 16, 25
![Page 81: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/81.jpg)
(1+X)/(1-X)(1+X)/(1-X)3 3 outputs <1, 4, 9, ..>outputs <1, 4, 9, ..>
X(1+X)/(1-X)X(1+X)/(1-X)3 3 outputs <0, 1, 4, 9, ..>outputs <0, 1, 4, 9, ..>
The kThe kthth entry is k entry is k22
![Page 82: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/82.jpg)
X(1+X)/(1-X)X(1+X)/(1-X)3 3 = = k k22XXkk
What does X(1+X)/(1-X)What does X(1+X)/(1-X)44 do?do?
![Page 83: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/83.jpg)
X(1+X)/(1-X)X(1+X)/(1-X)44 expands to : expands to :
SSkk X Xkk
where Swhere Skk is the sum of the is the sum of the first k squaresfirst k squares
![Page 84: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/84.jpg)
Aha! Thus, if there is an Aha! Thus, if there is an alternative interpretation of alternative interpretation of
the kthe kthth coefficient of coefficient of X(1+X)/(1-X)X(1+X)/(1-X)44
we would have a new way we would have a new way to get a formula for the sum to get a formula for the sum
of the first k squares.of the first k squares.
![Page 85: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/85.jpg)
What is the coefficient of Xk in the expansion of:
( 1 + X + X2 + X3 + X4
+ . . . . )n ?
Each path in the choice tree for the cross terms has n choices of exponent e1, e2, . . . , en ¸ 0. Each exponent can be any natural number.
Coefficient of Xk is the number of non-negative solutions to:
e1 + e2 + . . . + en = k
![Page 86: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/86.jpg)
What is the coefficient of Xk in the expansion of:
( 1 + X + X2 + X3 + X4
+ . . . . )n ?
n
n -1
1k
![Page 87: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/87.jpg)
( 1 + X + X2 + X3 + X4
+ . . . . )n =
k
k 0
nX
n -1
11
1n
k
X
![Page 88: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/88.jpg)
Using pirates and gold we found that:
k
k 0
nX
n -1
11
1n
k
X
k
k 0
X3
4
31
1
k
X
THUS:
![Page 89: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/89.jpg)
Vector programs -> Polynomials-> Closed form expression
![Page 90: Fibonacci Numbers, Polynomial Coefficients, and Vector Programs](https://reader035.vdocuments.mx/reader035/viewer/2022062300/56649e055503460f94af153a/html5/thumbnails/90.jpg)
REFERENCES
Coxeter, H. S. M. ``The Golden Section, Phyllotaxis, and Wythoff's Game.'' Scripta Mathematica 19, 135-143, 1953.
"Recounting Fibonacci and Lucas Identities" by Arthur T. Benjamin and Jennifer J. Quinn, College Mathematics Journal, Vol. 30(5): 359--366, 1999.