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Let(1)(2)(3)(OEIS A104457), where is the golden ratio, and(4)(5)(OEIS A002390).Define the Fibonacci hyperbolic sine by(6)(7)(8)The function satisfies(9)and for ,(10)where is a Fibonacci number. For , 2, ..., the values are therefore 1, 3, 8, 21, 55, ... (OEIS A001906).Define the Fibonacci hyperbolic cosine by(11)(12)(13)This function satisfies(14)and for ,(15)where is a Fibonacci number. For , 2, ..., the values are therefore 2, 5, 13, 34, 89, ... (OEIS A001519).Similarly, the Fibonacci hyperbolic tangent is defined by(16)and for ,(17)For , 2, ..., the values are therefore 1/2, 3/5, 8/13, 21/34, 55/89, ... (OEIS A001906 and A001519).

The th central fibonomial coefficient is defined as(1)(2)where is a fibonomial coefficient, is a Fibonacci number, is the golden ratio, and is a q-Pochhammer symbol (E. W. Weisstein, Dec. 8, 2009).For , 2, ..., the first few are 1, 6, 60, 1820, 136136, ... (OEIS A003267).

The fibonomial coefficient (sometimes also called simply the Fibonacci coefficient) is defined by(1)where and is a Fibonacci number. This coefficient satisfies(2)for , where is a Lucas number.The triangle of fibonomial coefficients is given by(3)(OEIS A010048). may be called the central fibonomial coefficient by analogy with the central binomial coefficient.

There are several results known as the Morgado identity. The first is(1)where is a Fibonacci number and is a Lucas number (Morgado 1987, Dujella 1995).A second Morgado identity is satisfied by generalized Fibonacci numbers ,(2)where(3)(4)(Morgado 1987, Dujella 1996).

A number written as a sum of nonconsecutive Fibonaccinumbers,where are 0 or 1 andEvery positive integer can be written uniquelyin such a form.

The tribonacci numbers are a generalization of the Fibonacci numbers defined by , , , and the recurrence equation(1)for (e.g., Develin 2000). They represent the case of the Fibonacci n-step numbers.The first few terms using the above indexing convention for , 1, 2, ... are 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, ... (OEIS A000073; which however adopts the alternate indexing convention and ).The first few prime tribonacci numbers are 2, 7, 13, 149, 19341322569415713958901, ... (OEIS A092836), which have indices 3, 5, 6, 10, 86, 97, 214, 801, 4201, 18698, 96878, ... (OEIS A092835), and no others with (E. W. Weisstein, Mar. 21, 2009).Using Brown's criterion, it can be shown that the tribonacci numbers are complete; that is, every positive number can be written as the sum of distinct tribonacci numbers. Moreover, every positive number has a unique Zeckendorf-like expansion as the sum of distinct tribonacci numbers and that sum does..

The Jacobsthal numbers are the numbers obtained by the s in the Lucas sequence with and , corresponding to and . They and the Jacobsthal-Lucas numbers (the s) satisfy the recurrence relation(1)The Jacobsthal numbers satisfy and and are 0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, ... (OEIS A001045). The Jacobsthal-Lucas numbers satisfy and and are 2, 1, 5, 7, 17, 31, 65, 127, 257, 511, 1025, ... (OEIS A014551). The properties of these numbers are summarized in Horadam (1996).Microcontrollers (and other computers) use conditional instructions to change the flow of execution of a program. In addition to branch instructions, some microcontrollers use skip instructions which conditionally bypass the next instruction. This winds up being useful for one case out of the four possibilities on 2 bits, 3 cases on 3 bits, 5 cases on 4 bits, 11 on 5 bits, 21 on 6 bits, 43 on 7 bits, 85 on 8 bits, ..., which are exactly the Jacobsthal numbers (Hirst 2006).The Jacobsthal..

The tetranacci numbers are a generalization of the Fibonacci numbers defined by , , , , and the recurrence relation(1)for . They represent the case of the Fibonacci n-step numbers. The first few terms for , 1, ... are 0, 1, 1, 2, 4, 8, 15, 29, 56, 108, 208, ... (OEIS A000078).The first few prime tetranacci numbers have indices 3, 7, 11, 12, 36, 56, 401, 2707, 8417, 14096, 31561, 50696, 53192, 155182, ... (OEIS A104534), corresponding to 2, 29, 401, 773, 5350220959, ... (OEIS A104535), with no others for (E. W. Weisstein, Mar. 21, 2009).An exact expression for the th tetranacci number for can be given explicitly by(2)where the three additional terms are obtained by cyclically permuting , which are the four roots of the polynomial(3)Alternately,(4)This can be written in slightly more concise form as(5)where is the th root of the polynomial(6)and and are in the ordering of the Wolfram Language's Root object.The tetranacci numbers..

The hexanacci numbers are a generalization of the Fibonacci numbers defined by , , , , , , and the recurrence relation(1)for . They represent the case of the Fibonacci n-step numbers.The first few terms for , 2, ... are 1, 1, 2, 4, 8, 16, 32, 63, 125, 248, ... (OEIS A001592).An exact formula for the th hexanacci number can be given explicitly in terms of the six roots of(2)as(3)The ratio of adjacent terms tends to the positive root of , namely 1.98358284342... (OEIS A118427), sometimes called the hexanacci constant.

The limiting rabbit sequence written as a binary fraction (OEIS A005614), where denotes a binary number (a number in base-2). The decimal value is(1)(OEIS A014565).Amazingly, the rabbit constant is also given by the continued fraction [0; , , , , ...] = [2, 2, 4, 8, 32, 256, 8192, 2097152, 17179869184, ...] (OEIS A000301), where are Fibonacci numbers with taken as 0 (Gardner 1989, Schroeder 1991). Another amazing connection was discovered by S. Plouffe. Define the Beatty sequence by(2)where is the floor function and is the golden ratio. The first few terms are 1, 3, 4, 6, 8, 9, 11, ... (OEIS A000201). Then(3)This is a special case of the Devil's staircase function with .The irrationality measure of is (D. Terr, pers. comm., May 21, 2004).

A sequence which arises in the hypothetical reproduction of a population of rabbits. Let the substitution system map correspond to young rabbits growing old, and correspond to old rabbits producing young rabbits. Starting with 0 and iterating using string rewriting gives the terms 1, 10, 101, 10110, 10110101, 1011010110110, .... A recurrence plot of the limiting value of this sequence is illustrated above.Converted to decimal, this sequence gives 1, 2, 5, 22, 181, ... (OEIS A005203), with the th term given by the recurrence relationwith , , and the th Fibonacci number.The limiting sequence written as a binary fraction (OEIS A005614), where denotes a binary number (i.e., a number written in base 2, so or 1), is called the rabbit constant.

The heptanacci numbers are a generalization of the Fibonacci numbers defined by , , , , , , , and the recurrence relation(1)for . They represent the case of the Fibonacci n-step numbers.The first few terms for , 2, ... are 1, 1, 2, 4, 8, 16, 32, 64, 127, 253, ... (OEIS A066178).An exact formula for the th heptanacci number can be given explicitly in terms of the seven roots of(2)as(3)The ratio of adjacent terms tends to the real root of , namely 1.99196419660... (OEIS A118428), sometimes called the heptanacci constant.

Consider the Fibonacci-like recurrence(1)where , , and each sign is chosen independently and at random with probability 1/2. Surprisingly, Viswanath (2000) showed that(2)(OEIS A078416) with probability one. This constantis sometimes known as Viswanath's constant.Considering the more general recurrence(3)the limit(4)exists for almost all values of . The critical value such that is given by(5)(OEIS A118288) and is sometimes known as theEmbree-Trefethen constant.Since Fibonacci numbers can be computed as products of Fibonacci Q-matrices, this same constant arises in the iterated multiplication of certain pairs of random matrices (Bougerol and Lacrois 1985, pp. 11 and 157).

A generalization of the Fibonacci numbers defined by and the recurrence relation(1)These are the sums of elements on successive diagonals of a left-justified Pascal's triangle beginning in the leftmost column and moving in steps of up and 1 right. The case equals the usual Fibonacci number. These numbers satisfy the identities(2)(3)(4)(5)(Bicknell-Johnson and Spears 1996). For the special case ,(6)Bicknell-Johnson and Spears (1996) give many further identities.Horadam (1965) defined the generalized Fibonacci numbers as , where , , , and are integers, , , and for . They satisfy the identities(7)(8)(9)(10)where(11)(12)(Dujella 1996). The final above result is due to Morgado (1987) and is called themorgado identity.Another generalization of the Fibonacci numbers is denoted . Given and , define the generalized Fibonacci number by for ,(13)(14)(15)where the plus and minus signs alternate...

The fibonorial , also called the Fibonacci factorial, is defined aswhere is a Fibonacci number. For , 2, ..., the first few fibonorials are 1, 1, 2, 6, 30, 240, 3120, 65520, ... (OEIS A003266).The fibonorials are asymptotic towhere is the Fibonacci factorial constant and is the golden ratio.The first few values of such that is prime are given by 4, 5, 6, 7, 8, 14, 15, ... (OEIS A059709), with no others less than 500.The first few values of such that is prime are given by 1, 2, 3, 4, 5, 6, 7, 8, 22, 28, ... (OEIS A053408), with no others less than 500.

There are two identities known as Catalan's identity.The first iswhere is a Fibonacci number. Letting gives Cassini's Identity.The second is the trivariate identity on partition of cubes into a sum of three squares given by

For the th Fibonacci number,This identity was also discovered by Simson (Coxeter and Greitzer 1967, p. 41; Coxeter 1969, pp. 165-168; Wells 1986, p. 62). It is a special case of Catalan's identity with .

The pentanacci numbers are a generalization of the Fibonacci numbers defined by , , , , , and the recurrence relation(1)for . They represent the case of the Fibonacci n-step numbers.The first few terms for , 2, ... are 1, 1, 2, 4, 8, 16, 31, 61, 120, 236, ... (OEIS A001591).The ratio of adjacent terms tends to the real root of , namely 1.965948236645485... (OEIS A103814), sometimes called the pentanacci constant.An exact formula for the th pentanacci number can be given explicitly in terms of the five roots of(2)as(3)The pentanacci numbers have generating function(4)

Binet's formula is a special case of the Binet form with , corresponding to the th Fibonacci number,(1)(2)where is the golden ratio. It was derived by Binet in 1843, although the result was known to Euler, Daniel Bernoulli, and de Moivre more than a century earlier.

The Fibonacci factorial constant is the constant appearing in the asymptotic growth of the fibonorials (aka. Fibonacci factorials) . It is given by the infinite product(1)where(2)and is the golden ratio.It can be given in closed form by(3)(4)(5)(OEIS A062073), where is a q-Pochhammer symbol and is a Jacobi theta function.

The hexanacci constant is the limiting ratio of adjacent hexanaccinumbers. It is the algebraic number(1)(2)(OEIS A118427), where denotes a polynomial root.

The heptanacci constant is the limiting ratio of adjacent heptanaccinumbers. It is the algebraic number(1)(2)(OEIS A118428), where denotes a polynomial root.

The tribonacci constant is ratio to which adjacent tribonaccinumbers tend, and is given by(1)(2)(3)(OEIS A058265).The tribonacci constant satisfies the identities(4)(5)(6)(7)(P. Moses, pers. comm., Feb. 21, 2005).The tribonacci constant is extremely prominent in the properties of the snubcube.

The tetranacci constant is ratio to which adjacent tetranaccinumbers tend, and is given by(1)(2)(OEIS A086088), where denotes a polynomial root.The tetranacci constant satisfies the identity(3)

The pentanacci constant is the limiting ratio of adjacent pentanaccinumbers. It is the algebraic number(1)(2)(OEIS A103814), where denotes a polynomial root.

The first few prime Lucas numbers are 2, 3, 7, 11, 29, 47, 199, 521, 2207, 3571, ... (OEIS A005479), corresponding to indices , 2, 4, 5, 7, 8, 11, 13, 16, 17, 19, 31, 37, 41, 47, 53, 61, 71, 79, 113, 313, 353, 503, 613, 617, 863, 1097, 1361, 4787, 4793, 5851, 7741, 8467, 10691, 12251, 13963, 14449, 19469, 35449, 36779, 44507, 51169, 56003, 81671, 89849, 94823, 140057, 148091, 159521, 183089, 193201, 202667, 344293, 387433, 443609, 532277, 574219, 616787, 631181, 637751, 651821, 692147, 901657, 1051849, ... (Dubner and Keller 1999, Lifchitz and Lifchitz; OEIS A001606). Only those up to index 56003 have been proven prime (Broadhurst and Irvine 2006; https://primes.utm.edu/primes/page.php?id=77992). As of Apr. 2009, the largest known Lucas probable prime is , which has decimal digits (R. Lifchitz, Mar. 2009)...

A Fibonacci prime is a Fibonacci number that is also a prime number. Every that is prime must have a prime index , with the exception of . However, the converse is not true (i.e., not every prime index gives a prime ).The first few (possibly probable) prime Fibonacci numbers are 2, 3, 5, 13, 89, 233, 1597, 28657, 514229, ... (OEIS A005478), corresponding to indices , 4, 5, 7, 11, 13, 17, 23, 29, 43, 47, 83, 131, 137, 359, 431, 433, 449, 509, 569, 571, 2971, 4723, 5387, ... (OEIS A001605). (Note that Gardner's statement that is prime (Gardner 1979, p. 161) is incorrect, especially since 531 is not even prime, which it must be for to be prime.) The following table summarizes Fibonacci (possibly probable) primes with index .termindexdigitsdiscovererstatus2453871126proven prime; https://primes.utm.edu/primes/page.php?id=511292593111946proven prime; https://primes.utm.edu/primes/page.php?id=374702696772023proven prime; https://primes.utm.edu/primes/page.php?id=3553727144313016proven..

Let be the th Fibonacci number. Then the sequence is complete, even if one is restricted to subsequences in which no two consecutive terms are both passed over (until the desired total is reached; Brown 1965, Honsberger 1985).

The sequence of Fibonacci numbers is periodic modulo any modulus (Wall 1960), and the period (mod ) is the known as the Pisano period (Wrench 1969). For , 2, ..., the values of are 1, 3, 8, 6, 20, 24, 16, 12, 24, 60, 10, ... (OEIS A001175).Since , the last digit of repeats with period 60, as first noted by Lagrange in 1774 (Livio 2002, p. 105). The last two digits repeat with a period of 300, and the last three with a period of 1500. In 1963, Geller found that the last four digits have a period of and the last five a period of . Jarden subsequently showed that for , the last digits have a period of (Livio 2002, pp. 105-106). The sequence of Pisano periods for , 10, 100, 1000, ... are therefore 60, 300, 1500, 15000, 150000, 1500000, ... (OEIS A096363). is even if (Wall 1960). iff for some integer (Fulton and Morris 1969, Wrench 1969)...

Closed forms are known for the sums of reciprocals of even-indexed Fibonaccinumbers(1)(2)(3)(4)(5)(6)(7)(OEIS A153386; Knopp 1990, Ch. 8, Ex. 114; Paszkowski 1997; Horadam 1988; Finch 2003, p. 358; E. Weisstein, Jan. 1, 2009; Arndt 2012), where is the golden ratio, is a q-polygamma function, and is a Lambert series (Borwein and Borwein 1987, pp. 91 and 95) and odd-indexed Fibonacci numbers(8)(9)(10)(11)(12)(13)(OEIS A153387; Landau 1899; Borwein and Borwein 1997, p. 94; E. Weisstein, Jan. 1, 2009; Arndt 2012), where is a Jacobi elliptic function. Together, these give a closed form for the reciprocal Fibonacci constant of(14)(15)(16)(17)(18)(OEIS A079586; Horadam 1988; Griffin 1992; Zhao 1999; Finch 2003, p. 358). The question of the irrationality of was formally raised by Paul Erdős and this sum was proved to be irrational by André-Jeannin (1989).Borwein..

Vorobiev's theorem states that if , then , where is a Fibonacci number and means divides . The theorem was discovered by Vorobiev in 1942, but not published until 1967. It was used by Y. Matiyasevich in his negative solution to the Hilbert's tenth problem.Note that the converse does not hold. For example, , but . The plot above shows values of for which and (black) and for which but (red).

The two recursive sequences(1)(2)with , and , , can be solved for the individual and . They are given by(3)(4)where(5)(6)(7)A useful related identity is(8)Binet's Fibonacci number formula is a special case of the Binet form for corresponding to .

The Lucas numbers are the sequence of integers defined by the linear recurrence equation(1)with and . The th Lucas number is implemented in the Wolfram Language as LucasL[n].The values of for , 2, ... are 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, ... (OEIS A000204).The Lucas numbers are also a Lucas sequence and are the companions to the Fibonacci numbers and satisfy the same recurrence.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 (Honsberger 1985, p. 122).The only square numbers in the Lucas sequence are 1 and 4 (Alfred 1964, Cohn 1964). The only triangular Lucas numbers are 1, 3, and 5778 (Ming 1991). The only cubic Lucas number is 1.Rather amazingly, if is prime, . The converse does not necessarily hold true, however, and composite numbers such that are known as Lucas pseudoprimes.For , 2, ..., the numbers of decimal digits in are..

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 WolframLanguage 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,..

A generalization of the Fibonacci numbers defined by the four constants and the definitions and together with the linear recurrence equationfor . With , , , and , the Horadam sequence reduces to the Fibonacci numbers.

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