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Below is a short version of the key items that need to fixed.
Pretty basic stuff as you see.
Should be a walk in the park for the Chadster.
The Fibonacci numbers are the sequence of numbers {F_n}_(n=1)^infty defined by the linear recurrence equation
F_n=F_(n-1)+F_(n-2)
(1)
with F_1=F_2=1. As a result of the definition (1), it is conventional to define F_0=0.
The Fibonacci numbers for n=1, 2, ... are 1, 1, 2, 3, 5, 8, 13, 21, ... (Sloane's A000045).
Fibonacci numbers can be viewed as a particular case of the Fibonacci polynomials F_n(x) with F_n=F_n(1).
Fibonacci numbers are implemented in Mathematica as Fibonacci[n].
The Fibonacci numbers are also a Lucas sequence U_n(1,-1), and are companions to the Lucas numbers (which satisfy the same recurrence equation).
A scrambled version 13, 3, 2, 21, 1, 1, 8, 5 (Sloane's A117540) of the first eight Fibonacci numbers appear as one of the clues left by murdered museum curator Jacque Sauniere in D. Brown's novel The Da Vinci Code (Brown 2003, pp. 43, 60-61, and 189-192). In the Season 1 episode "Sabotage" (2005) of the television crime drama NUMB3RS, math genius Charlie Eppes mentions that the Fibonacci numbers are found in the structure of crystals and the spiral of galaxies and a nautilus shell. In the Season 4 episode "Masterpiece" (2008) of the CBS-TV crime drama "Criminal Minds," the agents of the FBI Behavioral Analysis Unit are confronted by a serial killer who uses the Fibonacci sequence to determine the number of victims for each of his killing episodes. In this episode, character Dr. Reid also notices that locations of the killings lie on the graph of a golden spiral, and going to the center of the spiral allows Reid to determine the location of the killer's base of operations.
Binary plot of the Fibonacci sequence
The plot above shows the first 511 terms of the Fibonacci sequence represented in binary, revealing an interesting pattern of hollow and filled triangles (Pegg 2003). A fractal-like series of white triangles appears on the bottom edge, due in part to the fact that the binary representation of F_(2^n+2^(n+1)) ends in n+2 zeros. Many other similar properties exist.
The Fibonacci numbers give the number of pairs of rabbits n months after a single pair begins breeding (and newly born bunnies are assumed to begin breeding when they are two months old), as first described by Leonardo of Pisa (also known as Fibonacci) in his book Liber Abaci. Kepler also described the Fibonacci numbers (Kepler 1966; Wells 1986, pp. 61-62 and 65). Before Fibonacci wrote his work, the Fibonacci numbers had already been discussed by Indian scholars such as Gopāla (before 1135) and Hemachandra (c. 1150) who had long been interested in rhythmic patterns that are formed from one-beat and two-beat notes or syllables. The number of such rhythms having n beats altogether is F_(n+1), and hence these scholars both mentioned the numbers 1, 2, 3, 5, 8, 13, 21, ... explicitly (Knuth 1997, p. 80).
The numbers of Fibonacci numbers less than 10, 10^2, 10^3, ... are 6, 11, 16, 20, 25, 30, 35, 39, 44, ... (Sloane's A072353). For n=1, 2, ..., the numbers of decimal digits in F_(10^n) are 2, 21, 209, 2090, 20899, 208988, 2089877, 20898764, ... (Sloane's A068070). As can be seen, the initial strings of digits settle down to produce the number 208987640249978733769..., which corresponds to the decimal digits of lnphi=0.2089876... (Sloane's A097348), where phi is the golden ratio. This follows from the fact that for any power function f_n=c^n, the number of decimal digits for f_(10^n) is given by 10^nlog_(10)c.
The Fibonacci numbers F_n, are squareful for n=6, 12, 18, 24, 25, 30, 36, 42, 48, 50, 54, 56, 60, 66, ..., 372, 375, 378, 384, ... (Sloane's A037917) and squarefree for n=1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, ... (Sloane's A037918). 4|F_(6n) and 25|F_(25n) for all n, and there is at least one n<=2m such that m|F_n. No squareful Fibonacci numbers F_p are known with p prime.
The ratios of successive Fibonacci numbers F_n/F_(n-1) approaches the golden ratio phi as n approaches infinity, as first proved by Scottish mathematician Robert Simson in 1753 (Wells 1986, p. 62). The ratios of alternate Fibonacci numbers are given by the convergents to phi^(-2), where phi is the golden ratio, and are said to measure the fraction of a turn between successive leaves on the stalk of a plant (phyllotaxis): for elm and linden, 1/3 for beech and hazel, 2/5 for oak and apple, 3/8 for poplar and rose, 5/13 for willow and almond, etc. (Coxeter 1969, Ball and Coxeter 1987). The Fibonacci numbers are sometimes called pine cone numbers (Pappas 1989, p. 224). The role of the Fibonacci numbers in botany is sometimes called Ludwig's law (Szymkiewicz 1928; Wells 1986, p. 66; Steinhaus 1999, p. 299). However, botanist Cooke suggests caution in making correlations between botany and the Fibonacci sequence (Peterson 2006).
The equation (◇) is a linear recurrence equation
x_n=Ax_(n-1)+Bx_(n-2) n>=3,
(2)
so the closed form for F_n is given by
F_n=(alpha^n-beta^n)/(alpha-beta),
(3)
where alpha and beta are the roots of x^2=Ax+B. Here, A=B=1, so the equation becomes
x^2-x-1=0,
(4)
which has roots
x=1/2(1+/-sqrt(5)).
(5)
The closed form is therefore given by
F_n=((1+sqrt(5))^n-(1-sqrt(5))^n)/(2^nsqrt(5)).
(6)
This is known as Binet's Fibonacci number formula (Wells 1986, p. 62). Another closed form is
F_n = [1/(sqrt(5))((1+sqrt(5))/2)^n]
(7)
= [(phi^n)/(sqrt(5))],
(8)
where [x] is the nearest integer function (Wells 1986, p. 62).
Fibonacci
Using equation (7), the definition of F_n can be extended to negative integers n according to
F_(-n)=(-1)^(n+1)F_n.
(9)
More generally, the Fibonacci numbers can be extended to n a real number nu via
F_nu=1/(sqrt(5)){((1+sqrt(5))/2)^nu-(2/(1+sqrt(5)))^nucos(nupi)},
(10)
as plotted above.
FibonacciRoots
The Fibonacci function has zeros at x=0 and an infinite number of negative values that approach n+0.5 for all negative integers n, given by the solutions to
phi^(2x)=cos(pix),
(11)
where phi is the golden ratio. The first few roots are 0, x=-0.183802... (Sloane's A089260), -1.5707764..., -2.4704268..., -3.5108513..., ....
Another recurrence relation for the Fibonacci numbers is
F_(n+1)=|_(F_n(1+sqrt(5))+1)/2_|=|_phiF_n+1/2_|,
(12)
where |_x_| is the floor function and phi is the golden ratio. This expression follows from the more general recurrence relation
|F_(n+1) F_(n+2) ... F_(n+k); F_(n+k+1) F_(n+k+2) ... F_(n+2k); | | ... |; F_(n+k(k-1)+1) F_(n+k(k-1)+2) ... F_(n+k^2)|=0
(13)
for k>2. (The k=1 case is trivially F_(n+1), while the k=2 case is essentially Cassini's identity and therefore equal to (-1)^n.)
Another interesting determinant identity follows from defining A_n as the n
Pretty basic stuff as you see.
Should be a walk in the park for the Chadster.
The Fibonacci numbers are the sequence of numbers {F_n}_(n=1)^infty defined by the linear recurrence equation
F_n=F_(n-1)+F_(n-2)
(1)
with F_1=F_2=1. As a result of the definition (1), it is conventional to define F_0=0.
The Fibonacci numbers for n=1, 2, ... are 1, 1, 2, 3, 5, 8, 13, 21, ... (Sloane's A000045).
Fibonacci numbers can be viewed as a particular case of the Fibonacci polynomials F_n(x) with F_n=F_n(1).
Fibonacci numbers are implemented in Mathematica as Fibonacci[n].
The Fibonacci numbers are also a Lucas sequence U_n(1,-1), and are companions to the Lucas numbers (which satisfy the same recurrence equation).
A scrambled version 13, 3, 2, 21, 1, 1, 8, 5 (Sloane's A117540) of the first eight Fibonacci numbers appear as one of the clues left by murdered museum curator Jacque Sauniere in D. Brown's novel The Da Vinci Code (Brown 2003, pp. 43, 60-61, and 189-192). In the Season 1 episode "Sabotage" (2005) of the television crime drama NUMB3RS, math genius Charlie Eppes mentions that the Fibonacci numbers are found in the structure of crystals and the spiral of galaxies and a nautilus shell. In the Season 4 episode "Masterpiece" (2008) of the CBS-TV crime drama "Criminal Minds," the agents of the FBI Behavioral Analysis Unit are confronted by a serial killer who uses the Fibonacci sequence to determine the number of victims for each of his killing episodes. In this episode, character Dr. Reid also notices that locations of the killings lie on the graph of a golden spiral, and going to the center of the spiral allows Reid to determine the location of the killer's base of operations.
Binary plot of the Fibonacci sequence
The plot above shows the first 511 terms of the Fibonacci sequence represented in binary, revealing an interesting pattern of hollow and filled triangles (Pegg 2003). A fractal-like series of white triangles appears on the bottom edge, due in part to the fact that the binary representation of F_(2^n+2^(n+1)) ends in n+2 zeros. Many other similar properties exist.
The Fibonacci numbers give the number of pairs of rabbits n months after a single pair begins breeding (and newly born bunnies are assumed to begin breeding when they are two months old), as first described by Leonardo of Pisa (also known as Fibonacci) in his book Liber Abaci. Kepler also described the Fibonacci numbers (Kepler 1966; Wells 1986, pp. 61-62 and 65). Before Fibonacci wrote his work, the Fibonacci numbers had already been discussed by Indian scholars such as Gopāla (before 1135) and Hemachandra (c. 1150) who had long been interested in rhythmic patterns that are formed from one-beat and two-beat notes or syllables. The number of such rhythms having n beats altogether is F_(n+1), and hence these scholars both mentioned the numbers 1, 2, 3, 5, 8, 13, 21, ... explicitly (Knuth 1997, p. 80).
The numbers of Fibonacci numbers less than 10, 10^2, 10^3, ... are 6, 11, 16, 20, 25, 30, 35, 39, 44, ... (Sloane's A072353). For n=1, 2, ..., the numbers of decimal digits in F_(10^n) are 2, 21, 209, 2090, 20899, 208988, 2089877, 20898764, ... (Sloane's A068070). As can be seen, the initial strings of digits settle down to produce the number 208987640249978733769..., which corresponds to the decimal digits of lnphi=0.2089876... (Sloane's A097348), where phi is the golden ratio. This follows from the fact that for any power function f_n=c^n, the number of decimal digits for f_(10^n) is given by 10^nlog_(10)c.
The Fibonacci numbers F_n, are squareful for n=6, 12, 18, 24, 25, 30, 36, 42, 48, 50, 54, 56, 60, 66, ..., 372, 375, 378, 384, ... (Sloane's A037917) and squarefree for n=1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, ... (Sloane's A037918). 4|F_(6n) and 25|F_(25n) for all n, and there is at least one n<=2m such that m|F_n. No squareful Fibonacci numbers F_p are known with p prime.
The ratios of successive Fibonacci numbers F_n/F_(n-1) approaches the golden ratio phi as n approaches infinity, as first proved by Scottish mathematician Robert Simson in 1753 (Wells 1986, p. 62). The ratios of alternate Fibonacci numbers are given by the convergents to phi^(-2), where phi is the golden ratio, and are said to measure the fraction of a turn between successive leaves on the stalk of a plant (phyllotaxis): for elm and linden, 1/3 for beech and hazel, 2/5 for oak and apple, 3/8 for poplar and rose, 5/13 for willow and almond, etc. (Coxeter 1969, Ball and Coxeter 1987). The Fibonacci numbers are sometimes called pine cone numbers (Pappas 1989, p. 224). The role of the Fibonacci numbers in botany is sometimes called Ludwig's law (Szymkiewicz 1928; Wells 1986, p. 66; Steinhaus 1999, p. 299). However, botanist Cooke suggests caution in making correlations between botany and the Fibonacci sequence (Peterson 2006).
The equation (◇) is a linear recurrence equation
x_n=Ax_(n-1)+Bx_(n-2) n>=3,
(2)
so the closed form for F_n is given by
F_n=(alpha^n-beta^n)/(alpha-beta),
(3)
where alpha and beta are the roots of x^2=Ax+B. Here, A=B=1, so the equation becomes
x^2-x-1=0,
(4)
which has roots
x=1/2(1+/-sqrt(5)).
(5)
The closed form is therefore given by
F_n=((1+sqrt(5))^n-(1-sqrt(5))^n)/(2^nsqrt(5)).
(6)
This is known as Binet's Fibonacci number formula (Wells 1986, p. 62). Another closed form is
F_n = [1/(sqrt(5))((1+sqrt(5))/2)^n]
(7)
= [(phi^n)/(sqrt(5))],
(8)
where [x] is the nearest integer function (Wells 1986, p. 62).
Fibonacci
Using equation (7), the definition of F_n can be extended to negative integers n according to
F_(-n)=(-1)^(n+1)F_n.
(9)
More generally, the Fibonacci numbers can be extended to n a real number nu via
F_nu=1/(sqrt(5)){((1+sqrt(5))/2)^nu-(2/(1+sqrt(5)))^nucos(nupi)},
(10)
as plotted above.
FibonacciRoots
The Fibonacci function has zeros at x=0 and an infinite number of negative values that approach n+0.5 for all negative integers n, given by the solutions to
phi^(2x)=cos(pix),
(11)
where phi is the golden ratio. The first few roots are 0, x=-0.183802... (Sloane's A089260), -1.5707764..., -2.4704268..., -3.5108513..., ....
Another recurrence relation for the Fibonacci numbers is
F_(n+1)=|_(F_n(1+sqrt(5))+1)/2_|=|_phiF_n+1/2_|,
(12)
where |_x_| is the floor function and phi is the golden ratio. This expression follows from the more general recurrence relation
|F_(n+1) F_(n+2) ... F_(n+k); F_(n+k+1) F_(n+k+2) ... F_(n+2k); | | ... |; F_(n+k(k-1)+1) F_(n+k(k-1)+2) ... F_(n+k^2)|=0
(13)
for k>2. (The k=1 case is trivially F_(n+1), while the k=2 case is essentially Cassini's identity and therefore equal to (-1)^n.)
Another interesting determinant identity follows from defining A_n as the n
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