###### displaystyle

- For example, in this notation the sequence of even numbers could be written as ( 2 n ) n ∈ N {\displaystyle (2n)_{n\in \mathbb {N} }} . The sequence of squares could be written as ( n 2 ) n ∈ N {\displaystyle (n^{2})_{n\in \mathbb {N} }} . The variable n is called an index, and the set of values that it can take is called the index set. (wikipedia.org)
- In mathematics, the Lucas sequences U n ( P , Q ) {\displaystyle U_{n}(P,Q)} and V n ( P , Q ) {\displaystyle V_{n}(P,Q)} are certain constant-recursive integer sequences that satisfy the recurrence relation x n = P ⋅ x n − 1 − Q ⋅ x n − 2 {\displaystyle x_{n}=P\cdot x_{n-1}-Q\cdot x_{n-2}} where P {\displaystyle P} and Q {\displaystyle Q} are fixed integers. (wikipedia.org)
- Thus: a + b = P , {\displaystyle a+b=P\,,} a b = 1 4 ( P 2 − D ) = Q , {\displaystyle ab={\frac {1}{4}}(P^{2}-D)=Q\,,} a − b = D . {\displaystyle a-b={\sqrt {D}}\,.} Note that the sequence a n {\displaystyle a^{n}} and the sequence b n {\displaystyle b^{n}} also satisfy the recurrence relation. (wikipedia.org)
- The two Beatty sequences they generate, B r = ( ⌊ n r ⌋ ) n ≥ 1 {\displaystyle {\mathcal {B}}_{r}=(\lfloor nr\rfloor )_{n\geq 1}} and B s = ( ⌊ n s ⌋ ) n ≥ 1 {\displaystyle {\mathcal {B}}_{s}=(\lfloor ns\rfloor )_{n\geq 1}} , form a pair of complementary Beatty sequences. (wikipedia.org)
- 1 {\displaystyle s>1} so that the Beatty sequences B r {\displaystyle {\mathcal {B}}_{r}} and B s {\displaystyle {\mathcal {B}}_{s}} partition the set of positive integers: each positive integer belongs to exactly one of the two sequences. (wikipedia.org)
- In mathematics, a divisibility sequence is an integer sequence ( a n ) {\displaystyle (a_{n})} indexed by positive integers n such that if m ∣ n then a m ∣ a n {\displaystyle {\text{if }}m\mid n{\text{ then }}a_{m}\mid a_{n}} for all m, n. (wikipedia.org)
- Every sequence of the form a n = k n {\displaystyle a_{n}=kn} , for some nonzero integer k, is a divisibility sequence. (wikipedia.org)
- 0 {\displaystyle A>B>0} is a divisibility sequence. (wikipedia.org)
- Two important subgroups are the group of Appell sequences, which are those sequences for which the operator Q is mere differentiation, and the group of sequences of binomial type, which are those that satisfy the identity p n ( x + y ) = ∑ k = 0 n ( n k ) p k ( x ) p n − k ( y ) . {\displaystyle p_{n}(x+y)=\sum _{k=0}^{n}{n \choose k}p_{k}(x)p_{n-k}(y). (wikipedia.org)
- If sn(x) is a Sheffer sequence and pn(x) is the one sequence of binomial type that shares the same delta operator, then s n ( x + y ) = ∑ k = 0 n ( n k ) p k ( x ) s n − k ( y ) . {\displaystyle s_{n}(x+y)=\sum _{k=0}^{n}{n \choose k}p_{k}(x)s_{n-k}(y). (wikipedia.org)
- In particular, if { sn(x) } is an Appell sequence, then s n ( x + y ) = ∑ k = 0 n ( n k ) x k s n − k ( y ) . {\displaystyle s_{n}(x+y)=\sum _{k=0}^{n}{n \choose k}x^{k}s_{n-k}(y). (wikipedia.org)
- For example, the natural numbers N {\displaystyle \mathbb {N} } can be thought of as a lawlike sequence: the sequence can be fully constructively described by the unique element 0 and a successor function. (wikipedia.org)
- Similarly, a function f : N ↦ N {\displaystyle f:\mathbb {N} \mapsto \mathbb {N} } mapping from the natural numbers into the natural numbers effectively determines the value for any argument it takes, and thus describes a lawlike sequence. (wikipedia.org)
- Let α ∈ n {\displaystyle \alpha \in n} denote the relation "the sequence α {\displaystyle \alpha } begins with the initial sequence n {\displaystyle n} " for choice sequence α {\displaystyle \alpha } and finite segment n {\displaystyle n} (more specifically, n {\displaystyle n} will probably be an integer encoding a finite initial sequence). (wikipedia.org)
- thus, it must be the case that A {\displaystyle A} also holds for any lawless sequence β {\displaystyle \beta } sharing this initial sequence. (wikipedia.org)
- The axiom of density, given by: ∀ n ∃ α [ α ∈ n ] {\displaystyle \forall n\,\exists \alpha [\alpha \in n]} states that, for any finite prefix (encoded by) n {\displaystyle n} , there is some sequence α {\displaystyle \alpha } beginning with that prefix. (wikipedia.org)

###### mathematics

- In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed. (wikipedia.org)
- The study of sequence conservation overlaps with the fields of genomics, proteomics, evolutionary biology, phylogenetics, bioinformatics and mathematics. (wikipedia.org)
- In mathematics, a Beatty sequence (or homogeneous Beatty sequence) is the sequence of integers found by taking the floor of the positive multiples of a positive irrational number. (wikipedia.org)
- In intuitionistic mathematics, a choice sequence is a constructive formulation of a sequence. (wikipedia.org)
- Since the Intuitionistic school of mathematics, as formulated by L. E. J. Brouwer, rejects the idea of a completed infinity, in order to use a sequence (which is, in classical mathematics, an infinite object), we must have a formulation of a finite, constructible object that can serve the same purpose as a sequence. (wikipedia.org)

###### Encyclopedia of Intege

- For a large list of examples of integer sequences, see On-Line Encyclopedia of Integer Sequences. (wikipedia.org)
- The On-Line Encyclopedia of Integer Sequences. (wikipedia.org)
- A list of integer sequences related to the Toothpick Sequence from the On-line Encyclopedia of Integer Sequences. (wikipedia.org)

###### recurrence relation

- Sheffer sequences are thus examples of generalized Appell polynomials and hence have an associated recurrence relation. (wikipedia.org)

###### integers

- Rayleigh's theorem, named after Lord Rayleigh, states that the complement of a Beatty sequence, consisting of the positive integers that are not in the sequence, is itself a Beatty sequence generated by a different irrational number. (wikipedia.org)

###### Lucas

- Famous examples of Lucas sequences include the Fibonacci numbers, Mersenne numbers, Pell numbers, Lucas numbers, Jacobsthal numbers, and a superset of Fermat numbers. (wikipedia.org)
- Lucas sequences are named after the French mathematician Édouard Lucas. (wikipedia.org)
- More generally, Lucas sequences of the first kind are divisibility sequences. (wikipedia.org)

###### integer sequence

- The Fibonacci numbers are the integer sequence whose elements are the sum of the previous two elements. (wikipedia.org)

###### genomes

- Advances in high-throughput DNA sequencing and protein mass spectrometry has substantially increased the availability of protein sequences and whole genomes for comparison since the early 2000s. (wikipedia.org)
- Although insertion sequences are usually discussed in the context of prokaryotic genomes, certain eukaryotic DNA sequences belonging to the family of Tc1/mariner transposable elements may be considered to be, insertion sequences. (wikipedia.org)

###### OEIS

- sequence A003504 in the OEIS) The first non-integral value is x43. (wikipedia.org)
- sequence A108394 in the OEIS) Guy, Richard K. (1981). (wikipedia.org)
- note: IDs such as A139250 are IDs within the OEIS, and descriptions of the sequences can be located by entering these IDs in the OEIS search page. (wikipedia.org)
- sequence A000201 in the OEIS). (wikipedia.org)

###### Fibonacci

- for example, the nth element of the Fibonacci sequence is generally denoted Fn. (wikipedia.org)

###### finite

- Formally, a sequence can be defined as a function whose domain is either the set of the natural numbers (for infinite sequences) or the set of the first n natural numbers (for a sequence of finite length n). (wikipedia.org)
- at any point of execution of this procedure, we will have examined only a finite initial segment of the sequence. (wikipedia.org)

###### examples

- Other ways to denote a sequence are discussed after the examples. (wikipedia.org)
- Other examples of sequences include ones made up of rational numbers, real numbers, and complex numbers. (wikipedia.org)
- Examples of highly conserved sequences include the RNA components of ribosomes present in all domains of life, the homeobox sequences widespread amongst Eukaryotes, and the tmRNA in Bacteria. (wikipedia.org)
- are examples of Appell sequences. (wikipedia.org)

###### define

- These sequences define the optimal strategy for Wythoff's game, and are used in the definition of the Wythoff array As another example, for r = √2, we have s = 2 + √2. (wikipedia.org)

###### denote

- There are a number of ways to denote a sequence, some of which are more useful for specific types of sequences. (wikipedia.org)

###### Characteristics

- Characteristics of sequences: Two segments, usually no more than three or four Usually in only one direction: continuingly higher or lower Segments continue by same interval distance It is possible for melody or harmony to form a sequence without the other participating. (wikipedia.org)
- Insertion sequences have two major characteristics: they are small relative to other transposable elements (generally around 700 to 2500 bp in length) and only code for proteins implicated in the transposition activity (they are thus different from other transposons, which also carry accessory genes such as antibiotic resistance genes). (wikipedia.org)

###### include

- edit This table does not include uncommon and experimental sequences. (wikipedia.org)
- While spoiled gradient-echo sequences refer to a steady state of the longitudinal magnetization only, SSFP gradient-echo sequences include transverse coherences (magnetizations) from overlapping multi-order spin echoes and stimulated echoes. (wikipedia.org)

###### rational numbers

- In fact, every real number can be written as the limit of a sequence of rational numbers, e.g. via its decimal expansion. (wikipedia.org)

###### occur

- Conservation can occur in coding and non-coding nucleic acid sequences. (wikipedia.org)

###### irrational

- More generally, if p is a polynomial with at least one coefficient other than the constant term irrational then the sequence p(n) is uniformly distributed modulo 1. (wikipedia.org)

###### multiples

- For example, the drawings of a random variable uniform over a segment will be equidistributed in the segment, but there will be large gaps compared to a sequence which first enumerates multiples of ε in the segment, for some small ε, in an appropriately chosen way, and then continues to do this for smaller and smaller values of ε. (wikipedia.org)

###### species

- In evolutionary biology, conserved sequences are similar or identical sequences in nucleic acids (DNA and RNA) or proteins across species (orthologous sequences) or within a genome (paralogous sequences). (wikipedia.org)
- He incorporated powerful off-stage aliens to explain the story's titular artifact, and in pondering the backstory began to flesh out the basics of what would later become the main players and setting of the Sequence: a universe full of intelligent species that live in the shadow of the incomprehensible and god-like Xeelee. (wikipedia.org)

###### repetition

- The sequences of the gradient echo type allow to achieve very short repetition times, as the acquisition of an echo corresponds to the acquisition of a k-space line, and this acquisition can be made quick by increasing the amplitude of the gradients of rephasing and reading. (wikipedia.org)
- For comparison purposes, the repetition time of a gradient echo sequence is of the order of 3 milliseconds, versus about 30 ms of a spin echo sequence. (wikipedia.org)
- In general, SSFP MRI sequences are based on a (low flip angle) gradient-echo MRI sequence with a short repetition time which in its generic form has been described as the FLASH MRI technique. (wikipedia.org)
- The descending 5-6 sequences, also known as descending third sequences, consist of a series of chords whose root notes descend by a third each sequential repetition. (wikipedia.org)

###### given

- Thus, Brouwer formulated the choice sequence, which is given as a construction, rather than an abstract, infinite object. (wikipedia.org)

###### therefore

- It is designed to make the naming of DNA sequence features and variants consistent and therefore machine-readable and searchable. (wikipedia.org)
- The result is a bass line that moves down continuously stepwise, resulting in a figured bass of '5-6', and therefore, the standard descending 5-6 sequence. (wikipedia.org)
- For example, a1 = 1 says that 1 only occurs once in the sequence, so a2 cannot be 1 too, but it can be, and therefore must be, 2. (wikipedia.org)
- a1 = 1 Therefore, 1 occurs exactly one time in this sequence. (wikipedia.org)
- Therefore, if some sequence is equidistributed in [a, b], it is expected that this sequence can be used to calculate the integral of a Riemann-integrable function. (wikipedia.org)

###### commonly

- Descending fifths sequences, also known as "circle of fifths" sequences, are the most commonly used types of sequences, singular extended in some works of Claudio Monteverdi and Heinrich Schütz. (wikipedia.org)
- To this day, the Hubble sequence is the most commonly used system for classifying galaxies, both in professional astronomical research and in amateur astronomy. (wikipedia.org)

###### important

- In particular, sequences are the basis for series, which are important in differential equations and analysis. (wikipedia.org)
- Within a sequence, amino acids that are important for folding, structural stability, or that form a binding site may be more highly conserved. (wikipedia.org)
- Non-coding sequences important for gene regulation, such as the binding or recognition sites of ribosomes and transcription factors, may be conserved within a genome. (wikipedia.org)
- A gradient echo sequence is It is the base of many important derived sequences such as echo-planar imaging and SSFP stationary sequences. (wikipedia.org)
- An important subtype of the descending 5-6 sequence is the root position variant, also known as the Pachelbel sequence, due to the use of this sequence in Pachelbel's Canon. (wikipedia.org)

###### approaches

- Conserved sequences are typically identified by bioinformatics approaches based on sequence alignment. (wikipedia.org)

###### drawn

- Note that the first point above is slightly misleading, as we may specify, for example, that the values in a sequence be drawn exclusively from the set of natural numbers-we can specify, a priori, the range of the sequence. (wikipedia.org)
- On the left (in the sense that the sequence is usually drawn) lie the ellipticals. (wikipedia.org)

###### relation

- Sometimes the term Sheffer sequence is defined to mean a sequence that bears this relation to some sequence of binomial type. (wikipedia.org)

###### known

- Homology search tools may take an individual nucleic acid or protein sequence as input, or use statistical models generated from multiple sequence alignments of known related sequences. (wikipedia.org)
- At no point, then, is any particular future value of the sequence known. (wikipedia.org)
- An insertion sequence (also known as an IS, an insertion sequence element, or an IS element) is a short DNA sequence that acts as a simple transposable element. (wikipedia.org)

###### segment

- In a melody, a real sequence is a sequence where the subsequent segments are exact transpositions of the first segment, while a tonal sequence is a sequence where the subsequent segments are diatonic transpositions of the first. (wikipedia.org)
- Equidistribution is a rather weak criterion to express the fact that a sequence fills the segment leaving no gaps. (wikipedia.org)

###### particular

- A sequence can be thought of as a list of elements with a particular order. (wikipedia.org)
- In magnetic resonance imaging (MRI), a sequence is a particular setting of pulse sequences and pulsed field gradients, resulting in a particular image appearance. (wikipedia.org)
- There are two axioms in particular that we expect to hold of choice sequences as described above. (wikipedia.org)
- A particular insertion sequence may be named according to the form ISn, where n is a number (e.g. (wikipedia.org)

###### generally

- However, sequence conservation in ncRNAs is generally poor compared to protein-coding sequences, and base pairs that contribute to structure or function are often conserved instead. (wikipedia.org)
- The sequence is almost never unadorned as shown above, but is generally filled in with intervening chords. (wikipedia.org)

###### context

- The empty sequence ( ) is included in most notions of sequence, but may be excluded depending on the context. (wikipedia.org)

###### restrict

- The codon usage bias in some organisms may restrict the types of synonymous mutations in a sequence. (wikipedia.org)

###### Note

- From text to graphic organizer - A sequence organizer is used to simplify, in note form, events in a sequential order. (wikipedia.org)
- For example, if a descending fifths sequence in C major starts with the note C, the next note will be F, a perfect fifth below the first note. (wikipedia.org)

###### types

- There are many types of sequences, each with a unique pattern. (wikipedia.org)

###### tends

- The non-diatonic sequence tends to modulate to a new tonality or to cause temporarily tonicisation. (wikipedia.org)
- A sequence is thus equidistributed if the discrepancy DN tends to zero as N tends to infinity. (wikipedia.org)

###### form

- form a strong divisibility sequence. (wikipedia.org)

###### distinction

- On images of the brain, this sequence has a more pronounced distinction between gray matter (bright) and white matter (darker gray), but with little contrast between brain and CSF. (wikipedia.org)
- A distinction is made between lawless and lawlike sequences. (wikipedia.org)

###### often

- It is often useful to combine this notation with the technique of treating the elements of a sequence as variables. (wikipedia.org)
- The structure of stages in the toothpick sequence often resemble the T-square fractal, or the arrangement of cells in the Ulam-Warburton cellular automaton. (wikipedia.org)
- Christopher Mark (2006), Ritornellos and the amplification from melodies to Baroque lyrics are often built from sequences. (wikipedia.org)
- most of the luminosity is in the arms and not the bulge Although strictly part of the de Vaucouleurs system of classification, the Sd class is often included in the Hubble sequence. (wikipedia.org)

###### exactly

- Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. (wikipedia.org)
- Here, "complementary" means that every positive integer belongs to exactly one of these two sequences. (wikipedia.org)
- It follows that each coset of the group of Appell sequences contains exactly one sequence of binomial type. (wikipedia.org)
- 1 a2 = 2 2 occurs exactly 2 times in this sequence. (wikipedia.org)
- a4 = a5 = 3 4 occurs exactly 3 times in this sequence. (wikipedia.org)

###### However

- However these might not be integer sequences. (wikipedia.org)
- However, even earlier, in 1894 such sequences were briefly mentioned by John W. Strutt (3rd Baron Rayleigh) in the second edition of his book The Theory of Sound. (wikipedia.org)

###### different

- citation needed] In coding sequences, the nucleic acid and amino acid sequence may be conserved to different extents, as the degeneracy of the genetic code means that synonymous mutations in a coding sequence do not affect the amino acid sequence of its protein product. (wikipedia.org)

###### order

- In 2009, Baxter posted a detailed chronology of the Xeelee Sequence explaining the proper chronological reading order of all the novels, novellas, and short stories up to that year. (wikipedia.org)

###### usually

- The coding region in an insertion sequence is usually flanked by inverted repeats. (wikipedia.org)

###### series

- When students read a text that outlines a story, process or series of events, it can be useful to put the information in a sequence. (wikipedia.org)
- In August 2016, the entire series of all novels and stories (up to that date) was released as one volume in e-book format entitled Xeelee Sequence: The Complete Series. (wikipedia.org)
- Baxter's Destiny's Children series is part of the Xeelee Sequence. (wikipedia.org)
- Xeelee Sequence main novels: Destiny's Children sub-series novels: Collections of short stories and novellas: ^a Winner of the BSFA Award for Best Short Fiction, 2004. (wikipedia.org)
- The canonical example of a lawless sequence is the series of rolls of a die. (wikipedia.org)

###### segments

- In geometry, the toothpick sequence is a sequence of 2-dimensional patterns which can be formed by repeatedly adding line segments ("toothpicks") to the previous pattern in the sequence. (wikipedia.org)

###### time

- A highly conserved sequence is one that has remained relatively unchanged far back up the phylogenetic tree, and hence far back in geological time. (wikipedia.org)
- Over many generations, nucleic acid sequences in the genome of an evolutionary lineage can gradually change over time due to random mutations and deletions. (wikipedia.org)

###### whose

- Other notations can be useful for sequences whose pattern cannot be easily guessed, or for sequences that do not have a pattern such as the digits of π. (wikipedia.org)

###### contains

- A graphic organizer can be used as a teaching tool in two ways: From graphic organizer to text - A completed sequence organizer is used to create a piece of writing based on the information it contains. (wikipedia.org)

###### elements

- The number of elements (possibly infinite) is called the length of the sequence. (wikipedia.org)
- One way to specify a sequence is to list the elements. (wikipedia.org)
- Listing is most useful for infinite sequences with a pattern that can be easily discerned from the first few elements. (wikipedia.org)
- denotes the number of elements, out of the first n elements of the sequence, that are between c and d. (wikipedia.org)

###### thought

- Highly conserved DNA sequences are thought to have functional value, although the role for many highly conserved non-coding DNA sequences is poorly understood. (wikipedia.org)

###### values

- The concept can be generalized to sequences with values in any ring where the concept of divisibility is defined. (wikipedia.org)

###### constant

- Any constant sequence is a strong divisibility sequence. (wikipedia.org)

###### interval

- is a sequence contained in the interval [a, b]. (wikipedia.org)
- of real numbers is said to be equidistributed modulo 1 or uniformly distributed modulo 1 if the sequence of the fractional parts of an, denoted by {an} or by an−⌊an⌋, is equidistributed in the interval [0, (wikipedia.org)

###### real numbers

- rather, it is a determinate sequence of real numbers. (wikipedia.org)

###### major

- In the third and fourth bars of the "Air" from J.S. Bach's Orchestral Suite No. 3 in D major, the violin part forms a tonal sequence with the notes modified to fit the harmony, while the intervals in the bass line are unchanged, creating a real sequence. (wikipedia.org)

###### properties

- Sequences are useful in a number of mathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence properties of sequences. (wikipedia.org)

###### another

- Elliptic divisibility sequences are another class of such sequences. (wikipedia.org)
- Another axiom is required for lawless sequences. (wikipedia.org)
- Nevertheless, there exist another sort of transposons, called unit transposons, that do not carry insertion sequences at their extremities (e.g. (wikipedia.org)

###### case

- In the first case there is a spoiled sequence, such as the FLASH (Fast Low-Angle Shot) sequence, while in the second case there are SSFP (Steady-State Free Precession) sequences. (wikipedia.org)

###### pattern

- This sequence does not have any pattern that is easily discernible by eye, unlike the preceding sequence, which is increasing. (wikipedia.org)
- At least two instances of a sequential pattern-including the original statement-are required to identify a sequence, and the pattern should be based on several melody notes or at least two successive harmonies (chords). (wikipedia.org)
- The ascending fifths sequence, contrary to the descending fifths sequence, consists of a pattern of ascending fifths (or descending fourths). (wikipedia.org)

###### instead

- A sequence of the spin echo type must instead wait for the exhaustion of the signal that is formed spontaneously after the application of the excitation impulse before it can produce an echo (free induction decay). (wikipedia.org)

###### second

- Notice that any number in the first sequence is lacking in the second, and vice versa. (wikipedia.org)