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..
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, ...
If is a sequence of measurable functions, with for every , then
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 (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..
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..
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.
An infinite sequence of positive integers is called weakly independent if any relation with or and , except finitely often, implies for all .
A sequence defined from a finite sequence , , ..., by defining , where is the mex (minimum excluded value).
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.
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).
If , , , ... is an artistic sequence, then , , , ... is a melodic sequence. The recurrence relation obeyed by melodic series is
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.
A unimodal sequence is a finite sequence that first increases and then decreases. A sequence is unimodal if there exists a such thatand
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,..
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 .
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...
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..
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..
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.
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 .
The longest increasing scattered subsequence is the longest subsequence of increasing terms, where intervening nonincreasing terms may be dropped. Finding the largest scattered subsequence is a much harder problem. The longest increasing scattered subsequence of a partition can be found using LongestIncreasingSubsequence[p] in the Wolfram Language package Combinatorica` . For example, the longest increasing scattered subsequence of the permutation is , whereas the longest contiguous subsequence is .Any sequence of distinct integers must contain either an increasing or decreasing scattered subsequence of length (Erdős and Szekeres 1935; Skiena 1990, p. 75).
Given a sequence of real numbers , the supremum limit (also called the limit superior or upper limit), written and pronounced 'lim-soup,' is the limit ofas , where denotes the supremum. Note that, by definition, is nonincreasing and so either has a limit or tends to . For example, suppose , then for odd, , and for even, . Another example is , in which case is a constant sequence .When , the sequence converges to the real numberOtherwise, the sequence does not converge.
A finite sequence of real numbers is said to be logarithmically concave (or log-concave) ifholds for every with .A logarithmically concave sequence of positive numbers is also unimodal.If and are two positive log-concave sequences of the same length, then is also log-concave. In addition, if the polynomial has all its zeros real, then the sequence is log-concave (Levit and Mandrescu 2005).An example of a logarithmically concave sequence is the sequence of binomial coefficients for fixed and .
A polynomial is called logarithmically concave (or log-concave) if the sequence of its coefficients is logarithmically concave.If is log-convex and is unimodal, then is unimodal. However, the product of two log-convex polynomials is itself log-convex (Levit and Mandrescu 2005).
A sequence is said to be convergent if it approaches some limit(D'Angelo and West 2000, p. 259).Formally, a sequence converges to the limit if, for any , there exists an such that for . If does not converge, it is said to diverge. This condition can also be written asEvery bounded monotonic sequence converges.Every unbounded sequence diverges.
The inverse transformof the exponential transformwhich relate sequences , , ... and , , ....
The Connell sequence is the sequence obtained by starting with the first positive odd number (1), taking the next two even numbers (2, 4), the next three odd numbers (5, 7, 9), the next four even numbers (10, 12, 14, 16), and so on. The first few terms are 1, 2, 4, 5, 7, 9, 10, 12, 14, 16, 17, ... (OEIS A001614). A binary plot of the sequence from 1 to 255 is illustrated above.Amazingly, the terms of this sequence have the closed formThis shows immediately that
A sequence of numbers is complete if every positive integer is the sum of some subsequence of , i.e., there exist or 1 such that(Honsberger 1985, pp. 123-126). The Fibonacci numbers are complete. In fact, dropping one number still leaves a complete sequence, although dropping two numbers does not (Honsberger 1985, pp. 123 and 126). The sequence of primes with the element prepended,is complete, even if any number of primes each are dropped, as long as the dropped terms do not include two consecutive primes (Honsberger 1985, pp. 127-128). This is a consequence of Bertrand's postulate.
A subsequence of is a sequence defined by , where is an increasing sequence of indices (D'Angelo and West 2000).For example, the prime numbers are a subsequenceof the positive integers.Subsequence generation is implemented in the WolframLanguage as Subsequences.
A spectrum sequence is a sequence formed by successive multiples of a real number rounded down to the nearest integer . If is irrational, the spectrum is called a Beatty sequence.
Just as many interesting integer sequences can be defined and their properties studied, it is often of interest to additionally determine which of their elements are prime. The following table summarizes the indices of the largest known prime (or probable prime) members of a number of named sequences.sequenceOEISdigitsdiscoverersearch limitcommentsalternating factorialA00127259961260448M. Rodenkirch (Sep. 18, 2017)100000 (M. Rodenkirch, Dec. 15, 2017)finite sequence; largest certified prime has index 661; the rest are probable primesApéry-constant primeA119334141141E. W. Weisstein (May 14, 2006)9089 (E. W. Weisstein, Mar. 22, 2008)status unknownApéry number A092825662410136E. W. Weisstein (Mar. 2004) (E. W. Weisstein, Mar. 2004)probable primeApéry number 87E. W. Weisstein..
A sequence whose terms are integers. The most complete printed references for such sequences are Sloane (1973) and its update, Sloane and Plouffe (1995). Neil Sloane maintains the sequences from both these works in a vastly expanded on-line encyclopedia known as the On-Line Encyclopedia of Integer Sequences (https://www.research.att.com/~njas/sequences/). In this listing, sequences are identified by a unique 6-digit A-number. Sequences appearing in Sloane and Plouffe (1995) are ordered lexicographically and identified with a 4-digit M-number, and those appearing in Sloane (1973) are identified with a 4-digit N-number. To look up sequences by e-mail, send a message to either mailto:[email protected] or mailto:[email protected] containing lines of the form lookup 5 14 42 132 ... (note that spaces must be used instead of commas).Integer sequences can be analyzed by a variety of techniques (Sloane and Plouffe..
A sequence is called an infinitive sequence if, for every , for infinitely many . Write for the th index for which . Then as and range through , the array , called the associative array of , ranges through all of .
The Beatty sequence is a spectrum sequence with an irrational base. In other words, the Beatty sequence corresponding to an irrational number is given by , , , ..., where is the floor function. If and are positive irrational numbers such thatthen the Beatty sequences , , ... and , , ... together contain all the positive integers without repetition.The sequences for particular values of and are given in the following table (Sprague 1963; Wells 1986, pp. 35 and 40), where is the golden ratio.parameterOEISsequenceA0019511, 2, 4, 5, 7, 8, 9, 11, 12, ...A0019523, 6, 10, 13, 17, 20, 23, 27, 30, ...A0228381, 3, 5, 6, 8, 10, 12, 13, 15, 17, ...A0544062, 4, 7, 9, 11, 14, 16, 18, 21, 23, 26, ...A0228432, 5, 8, 10, 13, 16, 19, 21, 24, 27, 29, ...A0543851, 3, 4, 6, 7, 9, 11, 12, 14, 15, 17, 18, ...A0228443, 6, 9, 12, 15, 18, 21, 25, 28, 31, 34, ...A0543861, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, ...A0002011, 3, 4, 6, 8, 9, 11, 12, 14, 16, 17, 19, 21, ...A0019502, 5, 7, 10, 13,..
For a sequence , if for , then is increasing for . Conversely, if for , then is decreasing for .If and for all , then is increasing for . Conversely, if and for all , then is decreasing for .
The Banach density of a set of integers is defined asif the limit exists. If the is replaced with or , then the result is known as the upper or lower Banach density, respectively.In the ergodic theory approach to Szemerédi's theorem, Banach density must be used. (Although the statements of Szemerédi's theorem with different types of density are equivalent, the proofs are not easily converted from one density type to the other.)
There are at least two sequences attributed to B. Recamán. One is the sequence formed by taking and letting(1)which can be succinctly defined as "subtract if you can, otherwise add." The first few terms are 1, 3, 6, 2, 7, 13, 20, 12, 21, 11, ... (OEIS A005132), illustrated above.A view of the first 256 terms as binary bits is shown above.The terms 1, 2, 3, ... occur at positions 1, 4, 2, 131, 129, 3, 5, ... (OEIS A057167). The high-water marks in this sequence are 1, 4, 131, 99734, 181653, 328002, ... (OEIS A064227), which occur at positions 1, 2, 4, 19, 61, 879, ... (OEIS A064228).Another sequence defined by Recamán is the sequence obtained by letting and defining(2)(Guy and Nowakowski 1995, Sloane 1999). The first few terms of this sequence are1, 1, 2, 6, 24, 120, 20, 140, 1120, ... (OEIS A008336)...
Let and and for , let be the least integer which can be expressed as the sum of two or more consecutive terms. The resulting sequence is 1, 2, 3, 5, 6, 8, 10, 11, 14, 16, ... (OEIS A005243). Let and , form all possible expressions of the form for , and append them. The resulting sequence is 2, 3, 5, 9, 14, 17, 26, 27, ... (OEIS A005244).
A sequence in which no term divides any other. Let be the set , then the number of primitive subsets of are 2, 3, 5, 7, 13, 17, 33, 45, 73, 103, 205, 253, ... (OEIS A051026). For example, the five primitive sequences in are , , , , , , and .
A primefree sequence is sequence whose terms are never prime. Graham (1964) proved that there exist relatively prime positive integers and such that the recurrence equation(1)with and contains no prime numbers.In addition, Graham (1964) constructed a pair of numbers (one 33 digits and the other 34)(2)(3)satisfying this condition. Knuth (1990) subsequently found a 17-digit pair(4)(5)satisfying the same conditions. Almost immediately, Wilf (1990) found a smaller pair (one 17 digits and the other 16)(6)(7)Note that Hoffman (1998, p. 159) inadvertently inverted the order of the Wilf (1990) pair, thus obtaining a sequence that has prime terms for , 163, 190, 523, 1855, 3228, 3579, 6468, 7170, 10230, 12783, 17259, 60139, 91315, 97923, 101823, 156075, 182220, ... (OEIS A108156), with no others for (E. W. Weisstein, May 5, 2006).Nicol (1999) subsequently found the 12-digit pair ...
A sequence is called an infinitive sequence if, for every , for infinitely many . Write for the th index for which . Then as and range through , the array , called the associative array of , ranges through all of .
A series is called artistic if every three consecutiveterms have a common three-way ratioA series is also artistic iff its series bias is a constant. A geometric series with ratio is an artistic series with
An addition chain for a number is a sequence , such that each member after is the sum of two earlier (not necessarily distinct) ones. The number is called the length of the addition chain. For example,is an addition chain for 14 of length (Guy 1994).
A sequence is said to be periodic with period with if it satisfies for , 2, .... For example, is a periodic sequence with least period 2.
A set of residues (mod ) such that every nonzero residue can be uniquely expressed in the form . Examples include (mod 7) and (mod 13). A necessary condition for a difference set to exist is that be of the form . A sufficient condition is that be a prime power. Perfect sets can be used in the construction of perfect rulers.
A sequence of 0s and 1s is called an odd sequence if each of the sums for , 1, ..., is odd.
A set in which no element divides the sum of any nonempty subset of the other elements. For example, is dividing, since (and ), but is nondividing since 4 divides none of , and similarly for 6 and 7. The empty set and sets of length one are therefore trivially nondividing. Also, any set other than which contains 1 is dividing.Consider all possible subsets on the integers . Then the numbers of nondividing subsets on , , ... are 1, 2, 3, 5, 7, 11, 14, 21, ... (OEIS A051014). For example, the 11 nondividing sets in are , , , , , , , , , , , , , and .
A sequence of positive integers(1)is a nonaveraging sequence if it contains no three terms which are in an arithmeticprogression, i.e., terms such that(2)for distinct , , . The empty set and sets of length one are therefore trivially nonaveraging.Consider all possible subsets on the integers . There is one nonaveraging sequence on (), two on ( and ), four on , and so on. For example, 13 of the 16 subjects of are nonaveraging, with , , and excluded. The numbers of nonaveraging subsets on , , ... are 1, 2, 4, 7, 13, 23, 40, ... (OEIS A051013).Wróblewski (1984) showed that for infinite nonaveraging sequences,(3)
The grid shading problem is the problem of proving the unimodality of the sequence , where for fixed and , is the number of partitions of with at most parts and largest part at most . The grid shading problem was solved by Sylvester (1878) using invariant theory (Proctor 1982). Proctor (1982) gave the first elementary proof of this result.The q-binomial coefficients givethe generating function for this sequence.
A sequence is equidistributed ifffor each , 2, .... A consequence of this result is that the sequence is equidistributed, and hence dense, in the interval for irrational , where , 2, ... and is the fractional part of (Finch 2003).
A sequence of positive integers such that is irrational for all integer sequences . Erdős showed that (OEIS A001146) is an irrationality sequence.
The longest increasing (contiguous) subsequence of a given sequence is the subsequence of increasing terms containing the largest number of elements. For example, the longest increasing subsequence of the permutation is .It can be coded in the Wolfram Languageas follows. <<Combintorica` LongestContinguousIncreasingSubsequence[p_] := Last[ Split[Sort[Runs[p]], Length[#1] >= Length[#2]&] ]
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.
An infinite sequence of positive integers is called strongly independent if any relation , with , , or and except finitely often, implies for all .
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..
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 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)
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.
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)
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 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;..
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)
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.
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.
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).
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..
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.
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.
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..
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 .
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).
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...
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.
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 .