# Sequences

## Sequences Topics

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### Kolakoski sequence

The self-describing sequence consisting of "blocks" of single and double 1s and 2s, where a "block" is a single digit or pair of digits that is different from the digit (or pair of digits) in the preceding block. To construct the sequence, start with the single digit 1 (the first "block"). Here, the single 1 means that block of length one follows the first block. Therefore, require that the next block is 2, giving the sequence 12.Now, the 2 means that the next (third) block will have length two, so append 11 and obtain the sequence 1211. We have added two 1s, so the fourth and fifth blocks have length one each, giving 12112 and then 121121. As a result of adding 21, we obtain 121121221. As a result of adding 221, we obtain 12112122122112, and so on, giving the sequence 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, ... (OEIS A006928). The sequence after successive iterations is given by 1, 12, 1211, 121121, 121121221, ..., and the lengths..

### Nonarithmetic progression sequence

Given two starting numbers , the following table gives the unique sequences that contain no three-term arithmetic progressions.SloanesequenceA0032781, 2, 4, 5, 10, 11, 13, 14, 28, 29, 31, 32, ...A0331561, 3, 4, 6, 10, 12, 13, 15, 28, 30, 31, 33, ...A0331571, 4, 5, 8, 10, 13, 14, 17, 28, 31, 32, 35, ...A0331581, 5, 6, 8, 12, 13, 17, 24, 27, 32, 34, 38, ...A0331592, 3, 5, 6, 11, 12, 14, 15, 29, 30, 32, 33, ...A0331602, 4, 5, 7, 11, 13, 14, 16, 29, 31, 32, 34, ...A0331612, 5, 6, 9, 11, 14, 15, 18, 29, 32, 33, 36, ...A0331623, 4, 6, 7, 12, 13, 15, 16, 30, 31, 33, 34, ...A0331633, 5, 6, 8, 12, 14, 15, 17, 30, 32, 33, 35, ...A0331644, 5, 7, 8, 13, 14, 16, 17, 31, 32, 34, 35, ...

### Monotone convergence theorem

If is a sequence of measurable functions, with for every , then

### Giuga sequence

A finite, increasing sequence of integers such thatA sequence is a Giuga sequence iff it satisfiesfor , ..., . There are no Giuga sequences of length 2, one of length 3 (), two of length 4 ( and ), 3 of length 5 (, , and ), 17 of length 6, 27 of length 7, and hundreds of length 8. There are infinitely many Giuga sequences. It is possible to generate longer Giuga sequences from shorter ones satisfying certain properties.

### Wolfram sequences

Wolfram (2002, p. 123) considered the sequence related to the Collatzproblem obtained by iterating(1)starting with . This gives the sequence 1, 3, 6, 9, 15, 24, 36, 54, 81, 123, ... (OEIS A070885). The first 40 iterations are illustrated above, with each row being one iteration and the number obtained in that iteration represented in binary.Another set of sequences are given by(2)starting with various initial values . Interestingly, while taking , 2, 3, 4, 5, 7, 9, 10, ... give simple periodic sequences, the cases , 8, give complicated aperiodic sequences. 100 iterations starting at each of to 10 are illustrated above.Wolfram also considered the sequence 1, 1, 3, 3, 3, 5, 3, ... (OEIS A070864) defined by and(3)(Wolfram 2002, p. 129, (b)), the sequence 1, 1, 2, 2, 2, 4, 3, 4, 4, 4, ... (OEIS A070867) defined by and(4)(Wolfram 2002, p. 129, (f)), and the sequence 1, 1, 2, 2, 2, 3, 3, 4, 3, 4, ... (OEIS A070868) defined by and(5)(Wolfram..

### M&ouml;bius transform

The transformation of a sequence , , ... with(1)into the sequence , , ... via the Möbius inversion formula,(2)The transformation of to is sometimes called the sum-of-divisors transform. Two other equivalent formulations are given by(3)the right side of which is called a Lambert series,and(4)where is the Riemann zeta function (Sloane and Plouffe 1995, p. 21).Example Möbius transformations (Sloane and Plouffe 1995, p. 22) include for all , giving the inverse transform as , 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, ... (OEIS A000005), the divisor function of . The Möbius transform of gives , 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 12, ... (OEIS A000010), the totient function of . The inverse Möbius transform of the sequence and gives , 4, 0, 4, 8, 0, 0, 4, 4, ... (OEIS A004018), the number of ways of writing as a sum of two squares. The inverse Möbius transform of for prime and for composite gives the sequence , 1, 1, 1, 1, 2, 1, 1, 1, ... (OEIS..

### Geometric sequence

A geometric sequence is a sequence , , 1, ..., such that each term is given by a multiple of the previous one. Another equivalent definition is that a sequence is geometric iff it has a zero series bias. If the multiplier is , then the th term is given byTaking gives the simple special caseThe Season 1 episode "Identity Crisis" (2005) of the television crime drama NUMB3RS mentions geometric progressions.

### Weakly independent

An infinite sequence of positive integers is called weakly independent if any relation with or and , except finitely often, implies for all .

### Mex sequence

A sequence defined from a finite sequence , , ..., by defining , where is the mex (minimum excluded value).

### Fractal sequence

Given an infinitive sequence with associative array , then is said to be a fractal sequence 1. If , then there exists such that , 2. If , then, for every , there is exactly one such that . (As and range through , the array , called the associative array of , ranges through all of .) An example of a fractal sequence is 1, 1, 1, 1, 2, 1, 2, 1, 3, 2, 1, 3, 2, 1, 3, ....If is a fractal sequence, then the associated array is an interspersion. If is a fractal sequence, then the upper-trimmed subsequence is given by , and the lower-trimmed subsequence is another fractal sequence. The signature of an irrational number is a fractal sequence.

### Weakly complete sequence

A sequence of numbers is said to be weakly complete if every positive integer beyond a certain point is the sum of some subsequence of (Honsberger 1985). Dropping two terms from the Fibonacci numbers produces a sequence which is not even weakly complete. However, the sequenceis weakly complete, even with any finite subsequence deleted (Graham 1964).

### Melodic sequence

If , , , ... is an artistic sequence, then , , , ... is a melodic sequence. The recurrence relation obeyed by melodic series is

### Exponential transform

The exponential transform is the transformation of a sequence , , ... into a sequence , , ... according to the equationThe inverse ("logarithmic") transform is then given byThe exponential transform relates the number of labeled connected graphs on nodes satisfying some property with the corresponding total number (not necessarily connected) of labeled graphs on nodes. In this application, the transform is called Riddell's formula for labeled graphs.

### Unimodal sequence

A unimodal sequence is a finite sequence that first increases and then decreases. A sequence is unimodal if there exists a such thatand

### Ulam sequence

The Ulam sequence is defined by , , with the general term for given by the least integer expressible uniquely as the sum of two distinct earlier terms. The numbers so produced are sometimes called u-numbers or Ulam numbers.The first few numbers in the (1, 2)-Ulam sequence are 1, 2, 3, 4, 6, 8, 11, 13, 16, ... (OEIS A002858). Here, the first term after the initial (1, 2) is obviously 3 since . The next term is . (We don't have to worry about since it is a sum of a single term instead of distinct terms.) 5 is not a member of the sequence since it is representable in two ways, , but is a member.Proceeding in the manner, we can generate Ulam sequences for any , examples of which are given in the table below. Sloanesequence(1, 2)A0028581, 2, 3, 4, 6, 8, 11, 13, 16, 18, ...(1, 3)A0028591, 3, 4, 5, 6, 8, 10, 12, 17, 21, ...(1, 4)A0036661, 4, 5, 6, 7, 8, 10, 16, 18, 19, ...(1, 5)A0036671, 5, 6, 7, 8, 9, 10, 12, 20, 22, ...(2, 3)A0018572, 3, 5, 7, 8, 9, 13, 14, 18, 19, ...(2, 4)A0489512, 4,..

### Majorization

Let and be nonincreasing sequences of real numbers. Then majorizes if, for each , 2, ..., ,with equality if . Note that some caution is needed when consulting the literature, since the direction of the inequality is not consistent from reference to reference. An order-free characterization along the lines of Horn's theorem is also readily available. majorizes iff there exists a doubly stochastic matrix such that . Intuitively, if majorizes , then is more "mixed" than . Horn's theorem relates the eigenvalues of a Hermitian matrix to its diagonal entries using majorization. Given two vectors , then majorizes iff there exists a Hermitian matrix with eigenvalues and diagonal entries .

### Lucas sequence

Let , be integers satisfying(1)Then roots of(2)are(3)(4)so(5)(6)(7)(8)Now define(9)(10)for integer , so the first few values are(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)and(22)(23)(24)(25)(26)(27)(28)(29)(30)(31)(32)Closed forms for these are given by(33)(34)The sequences(35)(36)are called Lucas sequences, where the definition is usually extended to include(37)The following table summarizes special cases of and .Fibonacci numbersLucas numbersPell numbersPell-Lucas numbersJacobsthal numbersPell-Jacobsthal numbersThe Lucas sequences satisfy the general recurrencerelations(38)(39)(40)(41)(42)(43)Taking then gives(44)(45)Other identities include(46)(47)(48)(49)(50)These formulas allow calculations for large to be decomposed into a chain in which only four quantities must be kept track of at a time, and the number of steps needed is . The chain is particularly simple if has many 2s in its factorization...

### Ekg sequence

The EKG sequence is the integer sequence having 1 as its first term, 2 as its second, and with each succeeding term being the smallest number not already used that shares a factor with the preceding term. This results in the sequence 1, 2, 4, 6, 3, 9, 12, 8, 10, 5, 15, ... (OEIS A064413). When plotted as a connect-the-dots plot (left figure), the sequence looks somewhat like an electrocardiogram (abbreviated "EKG" in medical circles), so this sequence became known as the EKG sequence. Lagarias et al. have computed the first 10 million terms of the sequence (Lagarias et al. 2002, Peterson 2002).Every term appears exactly once in this sequence, and the primes occur in increasing order (Lagarias et al. 2002). The inverse permutation of the integers giving the sequence is 1, 2, 5, 3, 10, 4, 14, 8, 6, 9, 20, 7, 28, ... (OEIS A064664).Lagarias et al. (2002) established the boundsfor the term . For the first terms, whenever a prime occurs, it is immediately..

### Szemer&eacute;di's theorem

Szemerédi's theorem states that every sequence of integers that has positive upper Banach density contains arbitrarily long arithmetic progressions.A corollary states that, for any positive integer and positive real number , there exists a threshold number such that for every subset of with cardinal number larger than contains a -term arithmetic progression. van der Waerden's Theorem follows immediately by setting . The best bounds for van der Waerden numbers are derived from bounds for in Szemerédi's theorem.Szemerédi's theorem was conjectured by Erdős and Turán (1936). Roth (1953) proved the case , and was mentioned in his Fields Medal citation. Szemerédi (1969) proved the case , and the general theorem in 1975 as a consequence of Szemerédi's regularity lemma (Szemerédi 1975a), for which he collected a \$1000 prize from Erdos. Fürstenberg and Katznelson (1979) proved..

### Sylvester cyclotomic number

Given a Lucas sequence with parameters and , discriminant , and roots and , the Sylvester cyclotomic numbers are(1)where(2)is a primitive root of unity and the product is over all exponents relatively prime to such that .For small , the first few values are(3)(4)(5)(6)(7)(8)(9)These numbers satisfy(10)where as usual .Ward (1954) gave a primality test involving these numbers.

### Suzanne set

The th Suzanne set is defined as the set of composite numbers for which and , where(1)(2)and(3)(4)Every Suzanne set has an infinite number of elements.The following table gives the first few Suzanne numbers in for small .OEIS1A0182521, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, ...2A1022164, 8, 15, 22, 26, 35, 42, 44, 60, 62, 64, ...3A1022179, 24, 27, 42, 60, 72, 78, 81, 105, 114, ...The Suzanne set is a subset of the Monica set .

### Sturmian sequence

If a sequence has the property that the block growth function for all , then it is said to have minimal block growth, and the sequence is called a Sturmian sequence. An example of this is the sequence arising from the substitution system(1)(2)yielding , which gives us the Sturmian sequence 01001010....Sturm functions are sometimes also said to forma Sturmian sequence.

### Strongly independent

An infinite sequence of positive integers is called strongly independent if any relation , with , , or and except finitely often, implies for all .

### Collatz problem

A problem posed by L. Collatz in 1937, also called the mapping, problem, Hasse's algorithm, Kakutani's problem, Syracuse algorithm, Syracuse problem, Thwaites conjecture, and Ulam's problem (Lagarias 1985). Thwaites (1996) has offered a £1000 reward for resolving the conjecture. Let be an integer. Then one form of Collatz problem asks if iterating(1)always returns to 1 for positive . (If negative numbers are included, there are four known cycles (excluding the trivial 0 cycle): (4, 2, 1), (, ), (, , , , ), and (, , , , , , , , , , , , , , , , , ).)The members of the sequence produced by the Collatz are sometimes known as hailstone numbers. Conway proved that the original Collatz problem has no nontrivial cycles of length . Lagarias (1985) showed that there are no nontrivial cycles with length . Conway (1972) also proved that Collatz-type problems can be formally undecidable. Kurtz and Simon (2007) proved that a natural generalization of the..

### Stirling transform

The transformation of a sequence into a sequence by the formula(1)where is a Stirling number of the second kind. The inverse transform is given by(2)where is a Stirling number of the first kind (Sloane and Plouffe 1995, p. 23).The following table summarized Stirling transforms for some common sequences, where denotes the Iverson bracket and denotes the primes.OEIS1A0001101, 1, 2, 5, 15, 52, 203, ...A0054930, 1, 3, 10, 37, 151, 674, ...A0001101, 2, 5, 15, 52, 203, 877, ...A0855070, 0, 1, 4, 13, 41, 136, 505, ...A0244301, 0, 1, 3, 8, 25, 97, 434, 2095, ...A0244290, 1, 1, 2, 7, 27, 106, 443, ...A0339991, , 1, , 1, , ...Here, gives the Bell numbers. has the exponential generating function(3)

### Stern's diatomic series

Stern's diatomic series is the sequence(1)... (OEIS A002487) which arises in the Calkin-Wilftree. It is sometimes also known as the fusc function (Dijkstra 1982).The th term can be given by the recurrence equation(2)with and . A sum formula is given by(3)A generating function is given by(4)(5)

### Cauchy sequence

A sequence , , ... such that the metric satisfiesCauchy sequences in the rationals do not necessarily converge,but they do converge in the reals.Real numbers can be defined using either Dedekindcuts or Cauchy sequences.

### Lehmer number

A Lehmer number is a number generated by a generalization of a Lucas sequence. Let and be complex numbers with(1)(2)where and are relatively prime nonzero integers and is not a root of unity. Then the corresponding Lehmer numbers are(3)and the companion numbers(4)

### Casoratian

The Casoratian of sequences , , ..., is defined by the determinantThe Casoratian is implemented in the Wolfram Language as Casoratian[y1, y2, ..., n].The solutions , , ..., of the linear difference equationfor , 1, ..., are linearly independent sequences iff their Casoratian is nonzero for (Zwillinger 1995).

### Smarandache sequences

Smarandache sequences are any of a number of simply generated integer sequences resembling those considered in published works by Smarandache such as the consecutive number sequences and Euclid numbers (Iacobescu 1997). Some other "Smarandache" sequences are given below.1. The concatenation of copies of the integer : 1, 22, 333, 4444, 55555, ... (OEIS A000461; Marimutha 1997). For , they have the simple formula(1)where is a repunit. In general,(2)where is the number of digits in . Since the th term is always divisible by , numbers in this sequences can never be prime. 2. The concatenation of the first Fibonacci numbers: 1, 11, 112, 1123, 11235, ... (OEIS A019523; Marimutha 1997). 3. The smallest number that is the sum of squares of two distinct earlier terms: 1, 2, 5, 26, 29, 677, ... (OEIS A008318; Bencze 1997). 4. The smallest number that is the sum of squares of any number of distinct earlier terms: 1, 1, 2, 4, 5, 6, 16, 17, ... (OEIS A008319;..

### Legendre transform

The Legendre transform of a sequence is the sequence with terms given by(1)(2)where is a binomial coefficient (Jin and Dickinson 2000, Zudilin 2004). The inverse Legendre transform is then given by(3)where(4)(5)(Zudilin 2004).Strehl (1994) and Schmidt (1995) showed that(6)

### Carmichael sequence

A finite, increasing sequence of integers such thatfor , ..., , where indicates that divides . A Carmichael sequence has exclusive even or odd elements. There are infinitely many Carmichael sequences for every order.

### Silverman's sequence

Let , and let be the number of occurrences of in a nondecreasing sequence of integers. then the first few values of are 1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, ... (OEIS A001462). the asymptotic value of the th term is , where is the golden ratio.

### Brown's criterion

A sequence of nondecreasing positive integers is complete iff 1. . 2. For all , 3, ...,A corollary states that a sequence for which and is complete (Honsberger 1985).

### Signature sequence

Let be an irrational number, define , and let be the sequence obtained by arranging the elements of in increasing order. A sequence is said to be a signature sequence if there exists a positive irrational number such that , and is called the signature of .One can also define two extended signature sequences for positive rational by taking the in increasing order or decreasing order. These can be considered signature sequences for and , respectively, where is an infinitesimal.The signature of an irrational number or either signature of a rational number is a fractal sequence. Also, if is a signature or extended signature sequence, then the lower-trimmed subsequence is . It has been conjectured that every sequence with both of these properties is a signature or extended signature sequence.If every initial subsequence of a sequence is an initial subsequence of some signature sequence, then is either a signature sequence, an extended signature..

### Sequence dispersion

An array , of positive integers is called a dispersion if 1. The first column of is a strictly increasing sequence, and there exists a strictly increasing sequence such that 2. , 3. The complement of the set is the set , 4. for all for and for all for all . If an array is a dispersion, then it is an interspersion.

### Sequence density

Let a sequence be strictly increasing and composed of nonnegative integers. Call the number of terms not exceeding . Then the density is given by if the limit exists.

### Joyce sequence

The sequence of numbers giving the number of digits in the three-fold power tower . The values of for , 2, ... are 1, 16, 7625597484987, ... (OEIS A002488; Rossier 1948), so the Joyce sequence is 1, 2, 13, 155, 2185, 36306, ... (OEIS A054382). Laisant (1906) found the term , and Uhler (1947) published the logarithm of this number to 250 decimal places (Wells 1986, p. 208).The sequence is named in honor of the following excerpt from the "Ithaca" chapter of James Joyce's Ulysses: "Because some years previously in 1886 when occupied with the problem of the quadrature of the circle he had learned of the existence of a number computed to a relative degree of accuracy to be of such magnitude and of so many places, e.g., the 9th power of the 9th power of 9, that, the result having been obtained, 33 closely printed volumes of 1000 pages each of innumerable quires and reams of India paper would have to be requisitioned in order to contain the complete..

### Iteration sequence

A sequence of positive integers is called an iteration sequence if there exists a strictly increasing sequence of positive integers such that and for , 3, .... A necessary and sufficient condition for to be an iteration sequence isfor all .

### Borwein integrals

The Borwein integrals are the class of definiteintegrals defined byfor odd . The integrals are curious because the terms , 3, ..., 13 all have unit numerators, but , 17, ... do not. The sequence of values of for , 3, ... is given by 1/2, 1/6, 1/30, 1/210, 1/1890, 1/20790, 1/270270, 467807924713440738696537864469/1896516717212415135141110350293750000, ... (OEIS A068214 and A068215; Borwein et al. 2004, p. 98; Bailey et al. 2006).

### Binomial transform

The binomial transform takes the sequence , , , ... to the sequence , , , ... via the transformationThe inverse transform is(Sloane and Plouffe 1995, pp. 13 and 22). The inverse binomial transform of for prime and for composite is 0, 1, 3, 6, 11, 20, 37, 70, ... (OEIS A052467). The inverse binomial transform of for even and for odd is 0, 1, 2, 4, 8, 16, 32, 64, ... (OEIS A000079). Similarly, the inverse binomial transform of for odd and for even is 1, 2, 4, 8, 16, 32, 64, ... (OEIS A000079). The inverse binomial transform of the Bell numbers 1, 1, 2, 5, 15, 52, 203, ... (OEIS A000110) is a shifted version of the same numbers: 1, 2, 5, 15, 52, 203, ... (Bernstein and Sloane 1995, Sloane and Plouffe 1995, p. 22).The central and raw momentsof statistical distributions are also related by the binomial transform...

### Scholz conjecture

Let the minimal length of an addition chain for a number be denoted . Then the Scholz conjecture, also called the Scholz-Brauer conjecture or Brauer-Scholz conjecture, states thatThe conjecture has been proven for a variety of special cases but not in general.

### Binet forms

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 .

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