mathworld.wolfram.com Fibonacci Num

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Fibonacci Number -- from Wolfram MathWorld
Created, developed, and nurtured by Eric Weisstein at Wolfram Research


The Fibonacci numbers are the sequence of numbers defined by the linear recurrence equation
(1)
with . As a result of the definition (1), it is conventional to define .
The Fibonacci numbers for , 2, ... are 1, 1, 2, 3, 5, 8, 13, 21, ... (OEIS A000045).
Fibonacci numbers can be viewed as a particular case of the Fibonacci polynomials with .
Fibonacci numbers are implemented in the Wolfram Language as Fibonacci[n].
The Fibonacci numbers are also a Lucas sequence , and are companions to the Lucas numbers (which satisfy the same recurrence equation).

The above cartoon (Amend 2005) shows an unconventional sports application of the Fibonacci numbers (left two panels). (The right panel instead applies the Perrin sequence).
A scrambled version 13, 3, 2, 21, 1, 1, 8, 5 (OEIS A117540) of the first eight Fibonacci numbers appear as one of the clues left by murdered museum curator Jacque Saunière 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.

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 ends in zeros. Many other similar properties exist.
The Fibonacci numbers give the number of pairs of rabbits 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 beats altogether is , 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, , , ... are 6, 11, 16, 20, 25, 30, 35, 39, 44, ... (OEIS A072353). For , 2, ..., the numbers of decimal digits in are 2, 21, 209, 2090, 20899, 208988, 2089877, 20898764, ... (OEIS A068070). As can be seen, the initial strings of digits settle down to produce the number 208987640249978733769..., which corresponds to the decimal digits of (OEIS A097348), where is the golden ratio. This follows from the fact that for any power function , the number of decimal digits for is given by .
The Fibonacci numbers , are squareful for , 12, 18, 24, 25, 30, 36, 42, 48, 50, 54, 56, 60, 66, ..., 372, 375, 378, 384, ... (OEIS A037917) and squarefree for , 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, ... (OEIS A037918). and for all , and there is at least one such that . No squareful Fibonacci numbers are known with prime.
The ratios of successive Fibonacci numbers approaches the golden ratio as 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 , where 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
(2)
so the closed form for is given by
(3)
where and are the roots of . Here, , so the equation becomes
(4)
which has roots
(5)
The closed form is therefore given by
(6)
This is known as Binet's Fibonacci number formula (Wells 1986, p. 62). Another closed form is
where is the nearest integer function (Wells 1986, p. 62).

Using equation (7), the definition of can be extended to negative integers according to
(9)
More generally, the Fibonacci numbers can be extended to a real number via
(10)
as plotted above.

The Fibonacci function has zeros at and an infinite number of negative values that approach for all negative integers , given by the solutions to
(11)
where is the golden ratio. The first few roots are 0, (OEIS A089260), , , , ....
Another recurrence relation for the Fibonacci numbers is
(12)
where is the floor function and is the golden ratio. This expression follows from the more general recurrence relation
(13)
for . (The case is trivially , while the case is essentially Cassini's identity and therefore equal to .)
Another interesting determinant identity follows from defining as the matrix with zeros everywhere except and for (i.e., along the superdiagonal and subdiagonal). Then
(14)
(S. Markelov).
The generating function for the Fibonacci numbers is

By plugging in , this gives the curious addition tree illustrated above,
(18)
so
(19)
(Livio 2002, pp. 106-107).
The sum
(20)
(OEIS A079586) is known as the reciprocal Fibonacci constant.
Yuri Matiyasevich (1970) showed that there is a polynomial in , , and a number of other variables , , , ... having the property that iff there exist integers , , , ... such that . This led to the proof of the impossibility of the tenth of Hilbert's problems (does there exist a general method for solving Diophantine equations?) by Julia Robinson and Martin Davis in 1970 (Reid 1997, p. 107).

The Fibonacci number gives the number of ways for dominoes to cover a checkerboard, as illustrated in the diagrams above (Dickau).
The number of ways of picking a set (including the empty set) from the numbers 1, 2, ..., without picking two consecutive numbers is . The number of ways of picking a set (including the empty set) from the numbers 1, 2, ..., without picking two consecutive numbers (where 1 and are now consecutive) is , where is a Lucas number.
The probability of not getting two heads in a row in tosses of a coin is (Honsberger 1985, pp. 120-122). Fibonacci numbers are also related to the number of ways in which coin tosses can be made such that there are not three consecutive heads or tails. The number of ideals of an -element fence poset is the Fibonacci number .
Given a resistor network of 1- resistors, each incrementally connected in series or parallel to the preceding resistors, then the net resistance is a rational number having maximum possible denominator of .
The Fibonacci numbers are given in terms of the Chebyshev polynomial of the second kind by
(21)
Sum identities include
There are a number of particular pretty algebraic identities involving the Fibonacci numbers, including
(Brousseau 1972), Catalan's identity
(32)
d'Ocagne's identity
(33)
and the Gelin-Cesàro identity
(34)
Letting in (32) gives Cassini's identity
(35)
sometimes also called Simson's formula since it was also discovered by Simson (Coxeter and Greitzer 1967, p. 41; Coxeter 1969, pp. 165-168; Petkovšek et al. 1996, p. 12).
Johnson (2003) gives the very general identity
(36)
which holds for arbitrary integers , , , , and with and from which many other identities follow as special cases.
The Fibonacci numbers obey the negation formula
(37)
the addition formula
(38)
where is a Lucas number, the subtraction formula
(39)
the fundamental identity
(40)
conjugation relation
(41)
successor relation
(42)
double-angle formula
(43)
multiple-angle recurrence
(44)
multiple-angle formulas
(where (48) holds only for ), the extension
(50)
(A. Mihailovs, pers. comm., Jan. 24, 2003), product expansions
(51)
and
(52)
square expansion,
(53)
and power expansion
(54)
Honsberger (1985, p. 107) gives the general relations
In the case , then and for odd,
(58)
Similarly, for even,
(59)
Letting gives the identities

Sum formulas for include
(Wells 1986, p. 63), the latter of which shows that the shallow diagonals" of Pascal's triangle sum to Fibonacci numbers (Pappas 1989). Additional identities can be found throughout the Fibonacci Quarterly journal. A list of 47 generalized identities are given by Halton (1965).
In terms of the Lucas number ,
(Honsberger 1985, pp. 111-113). A remarkable identity is
(69)
(Honsberger 1985, pp. 118-119), which can be generalized to
(70)
(Johnson 2003). It is also true that
(71)
for odd, and
(72)
for even (Freitag 1996).
From (◇), the ratio of consecutive terms is
which is just the first few terms of the continued fraction for the golden ratio . Therefore,

ผลลัพธ์ (ไทย) 1:

mathworld.wolfram.com หมายเลขฟีโบนัชชี - จาก Wolfram แมธเวิลด์สร้าง พัฒนา และหล่อเลี้ยง โดย Eric Weisstein ที่วิจัย Wolfram หมายเลขฟีโบนัชชีเป็นลำดับของตัวเลขที่กำหนด ด้วยสมการเชิงเส้นเกิด (1) ด้วยการ จากคำนิยาม (1), มันเป็นปกติเพื่อกำหนด Fibonacci หมายเลขสำหรับ 2,...เป็น 1, 1, 2, 3, 5, 8, 13, 21, ... (OEIS A000045) สามารถดูหมายเลขฟีโบนัชชีเป็นกรณีเฉพาะของ polynomials ฟีโบนัชชีมี มีใช้หมายเลขฟีโบนัชชีในภาษา Wolfram เป็น Fibonacci [n] หมายเลขฟีโบนัชชีเป็นลำดับ Lucas และ สหายหมายเลข Lucas (ซึ่งเป็นไปตามสมการที่เกิดขึ้นเหมือนกัน) การ์ตูนข้างต้น (Amend 2005) แสดงประยุกต์กระเป๋ากีฬาของเลขฟีโบนัชชี (ซ้ายสองแผง) (แผงด้านขวาแทนใช้ลำดับ Perrin) A scrambled version 13, 3, 2, 21, 1, 1, 8, 5 (OEIS A117540) of the first eight Fibonacci numbers appear as one of the clues left by murdered museum curator Jacque Saunière 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. 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 ends in zeros. Many other similar properties exist. The Fibonacci numbers give the number of pairs of rabbits 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 beats altogether is , 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, , , ... are 6, 11, 16, 20, 25, 30, 35, 39, 44, ... (OEIS A072353). For , 2, ..., the numbers of decimal digits in are 2, 21, 209, 2090, 20899, 208988, 2089877, 20898764, ... (OEIS A068070). As can be seen, the initial strings of digits settle down to produce the number 208987640249978733769..., which corresponds to the decimal digits of (OEIS A097348), where is the golden ratio. This follows from the fact that for any power function , the number of decimal digits for is given by . The Fibonacci numbers , are squareful for , 12, 18, 24, 25, 30, 36, 42, 48, 50, 54, 56, 60, 66, ..., 372, 375, 378, 384, ... (OEIS A037917) and squarefree for , 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, ... (OEIS A037918). and for all , and there is at least one such that . No squareful Fibonacci numbers are known with prime. The ratios of successive Fibonacci numbers approaches the golden ratio as 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 , where 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 (2) so the closed form for is given by (3) where and are the roots of . Here, , so the equation becomes (4) which has roots (5) The closed form is therefore given by (6) This is known as Binet's Fibonacci number formula (Wells 1986, p. 62). Another closed form is where is the nearest integer function (Wells 1986, p. 62). Using equation (7), the definition of can be extended to negative integers according to (9) More generally, the Fibonacci numbers can be extended to a real number via (10) as plotted above. The Fibonacci function has zeros at and an infinite number of negative values that approach for all negative integers , given by the solutions to (11) where is the golden ratio. The first few roots are 0, (OEIS A089260), , , , .... Another recurrence relation for the Fibonacci numbers is (12) where is the floor function and is the golden ratio. This expression follows from the more general recurrence relation (13) for . (The case is trivially , while the case is essentially Cassini's identity and therefore equal to .) Another interesting determinant identity follows from defining as the matrix with zeros everywhere except and for (i.e., along the superdiagonal and subdiagonal). Then (14) (S. Markelov). The generating function for the Fibonacci numbers is By plugging in , this gives the curious addition tree illustrated above, (18) so (19) (Livio 2002, pp. 106-107). The sum (20) (OEIS A079586) is known as the reciprocal Fibonacci constant. Yuri Matiyasevich (1970) showed that there is a polynomial in , , and a number of other variables , , , ... having the property that iff there exist integers , , , ... such that . This led to the proof of the impossibility of the tenth of Hilbert's problems (does there exist a general method for solving Diophantine equations?) by Julia Robinson and Martin Davis in 1970 (Reid 1997, p. 107). The Fibonacci number gives the number of ways for dominoes to cover a checkerboard, as illustrated in the diagrams above (Dickau). The number of ways of picking a set (including the empty set) from the numbers 1, 2, ..., without picking two consecutive numbers is . The number of ways of picking a set (including the empty set) from the numbers 1, 2, ..., without picking two consecutive numbers (where 1 and are now consecutive) is , where is a Lucas number. The probability of not getting two heads in a row in tosses of a coin is (Honsberger 1985, pp. 120-122). Fibonacci numbers are also related to the number of ways in which coin tosses can be made such that there are not three consecutive heads or tails. The number of ideals of an -element fence poset is the Fibonacci number . Given a resistor network of 1- resistors, each incrementally connected in series or parallel to the preceding resistors, then the net resistance is a rational number having maximum possible denominator of . The Fibonacci numbers are given in terms of the Chebyshev polynomial of the second kind by (21) Sum identities include There are a number of particular pretty algebraic identities involving the Fibonacci numbers, including (Brousseau 1972), Catalan's identity (32) d'Ocagne's identity (33) and the Gelin-Cesàro identity (34) Letting in (32) gives Cassini's identity (35) sometimes also called Simson's formula since it was also discovered by Simson (Coxeter and Greitzer 1967, p. 41; Coxeter 1969, pp. 165-168; Petkovšek et al. 1996, p. 12). Johnson (2003) gives the very general identity (36) which holds for arbitrary integers , , , , and with and from which many other identities follow as special cases. The Fibonacci numbers obey the negation formula (37) the addition formula (38) where is a Lucas number, the subtraction formula (39) the fundamental identity (40) conjugation relation (41) successor relation (42) double-angle formula (43) multiple-angle recurrence (44) multiple-angle formulas (where (48) holds only for ), the extension (50) (A. Mihailovs, pers. comm., Jan. 24, 2003), product expansions (51) and (52) square expansion, (53) and power expansion (54) Honsberger (1985, p. 107) gives the general relations In the case , then and for odd, (58) Similarly, for even, (59) Letting gives the identities Sum formulas for include (Wells 1986, p. 63), the latter of which shows that the shallow diagonals" of Pascal's triangle sum to Fibonacci numbers (Pappas 1989). Additional identities can be found throughout the Fibonacci Quarterly journal. A list of 47 generalized identities are given by Halton (1965). In terms of the Lucas number , (Honsberger 1985, pp. 111-113). A remarkable identity is (69) (Honsberger 1985, pp. 118-119), which can be generalized to (70) (Johnson 2003). It is also true that (71) for odd, and (72) for even (Freitag 1996). From (◇), the ratio of consecutive terms is which is just the first few terms of the continued fraction for the golden ratio . Therefore,

ผลลัพธ์ (ไทย) 3:

mathworld.wolfram.com
Fibonacci หมายเลขจาก Wolfram แมธเวิลด์
สร้าง , พัฒนาและดูแลโดย อีริค weisstein ที่ Wolfram วิจัย


Fibonacci หมายเลขลำดับของตัวเลขที่กำหนด โดยการเชิงเส้นสมการ ( 1 )

ด้วย ผลของความหมาย ( 1 ) มันเป็นปกติที่จะกำหนด
Fibonacci หมายเลข 2 . . . . . . . 1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , . . . ( oeis a000045 )
ตัวเลข Fibonacci สามารถดูเป็นกรณีเฉพาะลำดับพหุนามด้วย .
ตัวเลข Fibonacci จะดำเนินการใน Wolfram ภาษาเป็นลำดับ [ Sample ]
ตัวเลข Fibonacci เป็นลูคัสลำดับและเป็นสหายกับลูคัส ตัวเลข ( ซึ่งสอดคล้องกับสมการการเดียวกัน )

การ์ตูนข้างต้น ( ปรับปรุง 2005 ) แสดงการแหกคอกกีฬาของตัวเลข Fibonacci ( เหลือ 2 แผง ) ( แผงด้านขวาแทนใช้แร็งลำดับ )
สัญญาณรบกวนรุ่น 13 , 3 , 2 , 2 , 1 , 1 , 8 , 5 ( oeis a117540 ) แรกของแปดตัวเลข Fibonacci ที่ปรากฏเป็นหนึ่งในปมที่ทิ้งไว้โดยฆ่าผู้ดูแลพิพิธภัณฑ์ชาร์ก sauni อีเบย์ในคือ สีน้ำตาลนวนิยายดาวินชีโค้ด ( สีน้ำตาลปี 2003. 43 , 60-61 และ 189-192 ) ในฤดูกาลที่ 1 ตอน " ก่อวินาศกรรม " ( 2005 ) ของอาชญากรรมละครเรื่อง Numb3rs คณิตศาสตร์อัจฉริยะ ชาร์ลี eppes กล่าวว่าตัวเลข Fibonacci ที่พบในโครงสร้างของผลึกและเกลียวกาแล็กซีและเปลือกหอย Nautilus . ในซีซั่น 4 ตอน " ชิ้นเอก " ( 2008 ) จากละครเรื่อง " อาชญากรรม cbs-tv จิตใจอาชญากร" ตัวแทนของหน่วยวิเคราะห์พฤติกรรมเอฟบีไอจะเผชิญหน้ากับฆาตกรต่อเนื่องที่ใช้ลำดับเลขเพื่อตรวจสอบจำนวนของเหยื่อที่เขาฆ่า แต่ละตอน ในตอนนี้ตัวละครด็อกเตอร์รี้ดยังสังเกตว่า ตำแหน่งของการฆ่าอยู่ในกราฟของเกลียวทอง