"Constant-recursive sequence In mathematics, a constant-recursive sequence or C-finite sequence is a sequence satisfying a linear recurrence with constant coefficients. An order-""d"" homogeneous linear recurrence with constant coefficients is an equation of the form where the ""d"" coefficients formula_2 are constants. A sequence formula_3 is a constant-recursive sequence if there is an order-""d"" homogeneous linear recurrence with constant coefficients that it satisfies for all formula_4. Equivalently, formula_5 is constant-recursive if the set of sequences is contained in a vector space whose dimension is finite. The sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... of"
"Fibonacci numbers in popular culture The Fibonacci numbers are a sequence of integers, starting with 0, 1 and continuing 1, 2, 3, 5, 8, 13, ..., each new number being the sum of the previous two. The Fibonacci numbers, and in conjunction the golden ratio, are a popular theme in culture. They have been mentioned in novels, films, television shows, and songs. The numbers have also been used in the creation of music, visual art, and architecture. John Waskom postulated that stages of human development followed the Fibonacci sequence, and that the unfolding psychology of human life would ideally be"
"thirteenth place, the value 233, though another manuscript carries it to the next place: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377. Fibonacci did not speak about the golden ratio as the limit of the ratio of consecutive numbers in this sequence. In the 19th century, a statue of Fibonacci was constructed and raised in Pisa. Today it is located in the western gallery of the Camposanto, historical cemetery on the Piazza dei Miracoli. There are many mathematical concepts named after Fibonacci because of a connection to the Fibonacci numbers. Examples include the Brahmagupta–Fibonacci"
"that every number in the sequence is the sum of the two previous numbers and can be found in the Mandelbrot Set. Starting with the period one cardioid and the period two bulb, the sequence of bulb periods where each successive bulb is the largest bulb attached to the cardioid between the previous two bulbs follows the Fibonacci sequence. For instance, the period three bulb is the next largest bulb after the period two bulb, the period five bulb is the largest between the period two and period three bulbs, and the period eight bulb is the largest between the"
"3, 5, 8, 13, …. treating the sequence as a message and each number as a symbol, there are almost as many symbols as there are characters in the message, giving an entropy of approximately . The first 128 symbols of the Fibonacci sequence has an entropy of approximately 7 bits/symbol, but the sequence can be expressed using a formula [ for , , ] and this formula has a much lower entropy and applies to any length of the Fibonacci sequence. In cryptanalysis, entropy is often roughly used as a measure of the unpredictability of a cryptographic key, though"
"sides of the preceding triangle in this series of triangles, and the shorter leg is equal to the difference between the preceding bypassed Fibonacci number and the shorter leg of the preceding triangle. The first triangle in this series has sides of length 5, 4, and 3. Skipping 8, the next triangle has sides of length 13, 12 (5 + 4 + 3), and 5 (8 − 3). Skipping 21, the next triangle has sides of length 34, 30 (13 + 12 + 5), and 16 (21 − 5). This series continues indefinitely. The triangle sides ""a"", ""b"", ""c"" can"
"notes in each of the five sections by the first five members of the Fibonacci series: 1, 2, 3, 5, and 8; that is, in the first section the notes appear in their original form, in section two each is divided in half, in section three into triplets, and so on (; ). Finally, a process of ""extreme compressions which begin to destroy the form to the point of its no longer being perceptible so as to bring about silence and motionless sound"" is imposed over the course of the work. ""The formula (whose skeleton is present in the first"
"Fibonacci number In mathematics, the Fibonacci numbers, commonly denoted form a sequence, called the Fibonacci sequence, such that each number is the sum of the two preceding ones, starting from 0 and 1. That is, and for . One has . In some books, and particularly in old ones, , the ""0"" is omitted, and the Fibonacci sequence starts with . The beginning of the Fibonacci sequence is thus: Fibonacci numbers are strongly related to the golden ratio: Binet's formula expresses the th Fibonacci number in terms of and the golden ratio, and imply that the ratio of two consecutive"
"two places before that. The first values are 1, 1, 1, 2, 3, 4, 6, 9, 13, 19, 28, 41, 60, 88, 129, 189, 277, 406, 595… (OEIS A000930). Many of the properties of the supergolden ratio are related to those of the golden ratio. For example, the n item of Narayana's sequence is the number of ways to tile a 1×n rectangle with 1×1 and 1×3 tiles, while the n term of the Fibonacci sequence is the number of ways to tile a 1×n rectangle with 1×1 and 1×2 tiles. φ−1=φ, and ψ−1=ψ. In Fibonacci's rabbit problem, each pair"
"1, 3, 21, 55 are the only triangular Fibonacci numbers, which was conjectured by Vern Hoggatt and proved by Luo Ming. With the exceptions of 1, 8 and 144 (""F"" = ""F"", ""F"" and ""F"") every Fibonacci number has a prime factor that is not a factor of any smaller Fibonacci number (Carmichael's theorem). As a result, 8 and 144 (""F"" and ""F"") are the only Fibonacci numbers that are the product of other Fibonacci numbers . The divisibility of Fibonacci numbers by a prime ""p"" is related to the Legendre symbol formula_105 which is evaluated as follows: If ""p"""
"Fibonacci numbers satisfies the recurrence with initial conditions Explicitly, the recurrence yields the values etc. The sequence 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, ... of Lucas numbers satisfies the same recurrence as the Fibonacci sequence but with initial conditions More generally, every Lucas sequence is a constant-recursive sequence. The geometric sequence formula_16 is constant-recursive, since it satisfies the recurrence formula_17 for all formula_18. A sequence that is eventually periodic with period length formula_19 is constant-recursive, since it satisfies formula_20 for all formula_4 for some ""d"". For any polynomial ""s""(""n""), the sequence of its values"
"of Fibonacci numbers modulo 3 begins: This sequence has period 8, so ""(3) = 8. With the exception of '(2) = 3, the Pisano period '(n) is always even. A simple proof of this can be given by observing that ""(n) is equal to the order of the Fibonacci matrix in the general linear group ""GL""(ℤ) of invertible 2 by 2 matrices in the finite ring ℤ of integers modulo n. Since Q has determinant -1, the determinant of Q is (-1), and since this must = 1 in ℤ, either n≤2 or """"(n) is even. If ""m"" and ""n"""
"2, or 4. Let ""p"" be the number after the first 0 after the combination 0, 1. Let the distance between the 0s be ""q"". For generalized Fibonacci sequences (satisfying the same recurrence relation, but with other initial values, e.g. the Lucas numbers) the number of occurrences of 0 per cycle is 0, 1, 2, or 4. The ratio of the Pisano period of ""n"" and the number of zeros modulo ""n"" in the cycle gives the ""rank of apparition"" or ""Fibonacci entry point"" of ""n"". That is, smallest index ""k"" such that ""n"" divides ""F""(""k""). They are: In Renault's"
"Maynard begins with the seventh number of the Fibonacci sequence (13), implying a missing verse in between. He descends back down with the following pattern; 13-8-5-3. ""As below so above and beyond I imagine (13). Drawn beyond the lines of reason (8). Push the envelope (5). Watch it bend (3)."" The second verse adds the missing line to complete the sequence; ""There is (2), so (1), much (1), more and (2), beckons me (3), to look through to these (5), infinite possibilities (8)."" 1-1-2-3-5-8-5-3-2-1-1-2-3-5-8-13-8-5-3. The song's introduction also ends at the 1 minute and 37 seconds mark, where the first"
"of the ""n""th month, the number of pairs of rabbits is equal to the number of new pairs (that is, the number of pairs in month ""n"" − 2) plus the number of pairs alive last month (that is, ""n"" − 1). This is the ""n""th Fibonacci number. The name ""Fibonacci sequence"" was first used by the 19th-century number theorist Édouard Lucas. The first 21 Fibonacci numbers ""F"" for ""n"" = 0, 1, 2, ..., 20 are: The sequence can also be extended to negative index ""n"" using the re-arranged recurrence relation Thus the bidirectional sequence is The Fibonacci numbers"
"and its reciprocal: The convergents of these continued fractions (1/1, 2/1, 3/2, 5/3, 8/5, 13/8, ..., or 1/1, 1/2, 2/3, 3/5, 5/8, 8/13, ...) are ratios of successive Fibonacci numbers. The equation = 1 + likewise produces the continued square root, or infinite surd, form: An infinite series can be derived to express phi: Also: These correspond to the fact that the length of the diagonal of a regular pentagon is times the length of its side, and similar relations in a pentagram. The number turns up frequently in geometry, particularly in figures with pentagonal symmetry. The length of a"
"from longest to shortest there are 1, 2, 3, 4, 3, and 2 variants (144, 89/91, 55/56/57, 34/35/36/37, 21/22/23, and 13/14). In order to achieve the specified numbers of moments, the variants of the shorter values are duplicated, again according to the Fibonacci series (13 × 5, 14 × 8; 21 × 3, 22 × 3, 23 × 2; 34 × 2, and the rest with single instances) . Each of the 32 moments is then subdivided into from two to thirteen subsections, again using Fibonacci numbers, in most cases with some values repeated. For example, one of the composer’s"
"loci. This process is called arrhenotokous parthenogenesis or simply arrhenotoky. Because the male bee technically has only a mother, and no father, its genealogical tree is unusual. The first generation has one member (the male). One generation back also has one member (the mother). Two generations back are two members (the mother and father of the mother). Three generations back are three members. Four back are five members. That is, the numbers in each generation going back are 1, 1, 2, 3, 5, 8, ... – the Fibonacci sequence. Much debate and controversy exist in the scientific literature about the"
"sequence: The n digit of the word is formula_16 where formula_14 is the golden ratio and formula_15 is the floor function. The word is related to the famous sequence of the same name (the Fibonacci sequence) in the sense that addition of integers in the inductive definition is replaced with string concatenation. This causes the length of ""S"" to be ""F"", the (""n"" + 2)th Fibonacci number. Also the number of 1s in ""S"" is ""F"" and the number of 0s in ""S"" is ""F"". Fibonacci based constructions are currently used to model physical systems with aperiodic order such as"
"is identifiable as a sequence of numbers in which any given number is the sum of the two numbers that precede it: 1, 1, 2, 3, 5, 8, etc., ad infinitum. From 1969 Merz employed the Fibonacci sequence in performances and installations throughout his career to represent the universal principles of creation and growth: climbing up the Guggenheim Museum in New York (1971) or the spire of a Turin landmark (1984), or perched in neon on a stack of newspapers among the old masters of Naples' Capodimonte Gallery (1987). In 1972 he illustrated the Fibonacci progression with a series of"
"the Pisano periods is an open problem, which includes as a subproblem a special instance of the problem of finding the multiplicative order of a modular integer or of an element in a finite field. However, for any particular ""n"", the Pisano period may be found as an instance of cycle detection. Starting with 5, every second Fibonacci number is the length of the hypotenuse of a right triangle with integer sides, or in other words, the largest number in a Pythagorean triple. The length of the longer leg of this triangle is equal to the sum of the three"
"It starts from 1, the next number is 1, and the next number being 2, creates the 2+1 which is 3, continuing in this mathematical progression. That’s how they found the chord progression. It began linking up to the Fibonacci sequence."" The syllables Maynard sings in the first verse follow the first six numbers in the pattern, ascending and descending in the sequence 1-1-2-3-5-8-5-3. ""Black (1), then (1), white are (2), all I see (3), in my infancy (5). Red and yellow then came to be (8), reaching out to me (5). Lets me see (3)."" In the next verse,"
"period three and period five bulbs. These correspond with the numbers of the Fibonacci sequence beginning with 1, 2, 3, 5, 8, 13, 21, and so on. The Mandelbrot set shows more intricate detail the closer one looks or magnifies the image, usually called ""zooming in"". The following example of an image sequence zooming to a selected ""c"" value gives an impression of the infinite richness of different geometrical structures and explains some of their typical rules. The magnification of the last image relative to the first one is about 10 to 1. Relating to an ordinary monitor, it represents"
"""n"" are: Pisano period In number theory, the ""n""th Pisano period, written π(""n""), is the period with which the sequence of Fibonacci numbers taken modulo ""n"" repeats. Pisano periods are named after Leonardo Pisano, better known as Fibonacci. The existence of periodic functions in Fibonacci numbers was noted by Joseph Louis Lagrange in 1774. The Fibonacci numbers are the numbers in the integer sequence: defined by the recurrence relation For any integer ""n"", the sequence of Fibonacci numbers ""F"" taken modulo ""n"" is periodic. The Pisano period, denoted """"(""n""), is the length of the period of this sequence. For example,"
"with durations of 144, 55 + 89, 144, and 34 + 21 + 34, + 55 units . These are then subdivided further into smaller measures. Each Fibonacci duration is associated throughout the composition with a particular articulation type. The value 13, for example, is associated with trills. However, when such a value is also a part of a larger one, e.g., when a section of 34 units is divided 13 + 21, the shorter section takes on in addition the character associated with 34, which is repetition of notes. On the other hand, where 13 combines with 8 to"
"terms of the floor function: Similarly, if we already know that the number ""F"" > 1 is a Fibonacci number, we can determine its index within the sequence by where formula_30 can be computed using logarithms to other usual bases. For example, formula_31. Johannes Kepler observed that the ratio of consecutive Fibonacci numbers converges. He wrote that ""as 5 is to 8 so is 8 to 13, practically, and as 8 is to 13, so is 13 to 21 almost"", and concluded that these ratios approach the golden ratio formula_32. This convergence holds regardless of the starting values, excluding 0"
"representation begins with the formula_16th Fibonacci number. Each Wythoff pair occurs exactly once in the Wythoff array, as a consecutive pair of numbers in the same row, with an odd index for the first number and an even index for the second. Because each positive integer occurs in exactly one Wythoff pair, each positive integer occurs exactly once in the array . Every sequence of positive integers satisfying the Fibonacci recurrence occurs, shifted by at most finitely many positions, in the Wythoff array. In particular, the Fibonacci sequence itself is the first row, and the sequence of Lucas numbers appears"
"Lifchitz in 2018. It was shown by Nick MacKinnon that the only Fibonacci numbers that are also members of the set of prime twins are 3, 5 and 13. A prime formula_3 divides formula_4 if and only if ""p"" is congruent to ±1 modulo 5, and ""p"" divides formula_5 if and only if is congruent to ±2 modulo 5. (For ""p"" = 5, ""F"" = 5 so 5 divides ""F"") Fibonacci numbers that have a prime index ""p"" do not share any common divisors greater than 1 with the preceding Fibonacci numbers, due to the identity: which implies the infinitude"
"Pisano period In number theory, the ""n""th Pisano period, written π(""n""), is the period with which the sequence of Fibonacci numbers taken modulo ""n"" repeats. Pisano periods are named after Leonardo Pisano, better known as Fibonacci. The existence of periodic functions in Fibonacci numbers was noted by Joseph Louis Lagrange in 1774. The Fibonacci numbers are the numbers in the integer sequence: defined by the recurrence relation For any integer ""n"", the sequence of Fibonacci numbers ""F"" taken modulo ""n"" is periodic. The Pisano period, denoted """"(""n""), is the length of the period of this sequence. For example, the sequence"
"be calculated directly: These formulas satisfy formula_125 for all ""n"", but they only represent triangle sides when ""n"" > 2. Any four consecutive Fibonacci numbers ""F"", ""F"", ""F"" and ""F"" can also be used to generate a Pythagorean triple in a different way: Since ""F"" is asymptotic to formula_127, the number of digits in ""F"" is asymptotic to formula_128. As a consequence, for every integer ""d"" > 1 there are either 4 or 5 Fibonacci numbers with ""d"" decimal digits. More generally, in the base ""b"" representation, the number of digits in ""F"" is asymptotic to formula_129 The Fibonacci numbers"
"puzzle ""What is the next number in the sequence 1, 11, 21, 1211, 111221?"" is sometimes referred to as the Cuckoo's Egg, from a description of Morris in Clifford Stoll's book ""The Cuckoo's Egg"". There are many possible variations on the rule used to generate the look-and-say sequence. For example, to form the ""pea pattern"" one reads the previous term and counts all instances of each digit, listed in order of their first appearance, not just those occurring in a consecutive block. Thus, beginning with the seed 1, the pea pattern proceeds 1, 11 (""one 1""), 21 (""two 1s""), 1211"
"element is the sum of the previous elements (with the exception of the first elements in the sequence). The usual Fibonacci numbers are a Fibonacci sequence of order 2. The cases and have been thoroughly investigated. The number of compositions of nonnegative integers into parts that are at most is a Fibonacci sequence of order . The sequence of the number of strings of 0s and 1s of length that contain at most consecutive 0s is also a Fibonacci sequence of order . These sequences, their limiting ratios, and the limit of these limiting ratios, were investigated by Mark Barr"
"The sequence also has a variety of relationships with the Fibonacci numbers, like the fact that adding any two Fibonacci numbers two terms apart in the Fibonacci sequence results in the Lucas number in between. Similar to the Fibonacci numbers, each Lucas number is defined to be the sum of its two immediate previous terms, thereby forming a Fibonacci integer sequence. The first two Lucas numbers are ""L"" = 2 and ""L"" = 1 as opposed to the first two Fibonacci numbers ""F"" = 0 and ""F"" = 1. Though closely related in definition, Lucas and Fibonacci numbers exhibit distinct"