Replacing the logistic equation(1)with the quadratic recurrence equation(2)where (sometimes also denoted ) is a positive constant sometimes known as the "biotic potential" gives the so-called logistic map. This quadratic map is capable of very complicated behavior. While John von Neumann had suggested using the logistic map as a random number generator in the late 1940s, it was not until work by W. Ricker in 1954 and detailed analytic studies of logistic maps beginning in the 1950s with Paul Stein and Stanislaw Ulam that the complicated properties of this type of map beyond simple oscillatory behavior were widely noted (Wolfram 2002, pp. 918-919).The first few iterations of the logistic map (2) give(3)(4)(5)where is the initial value, plotted above through five iterations (with increasing iteration number indicated by colors; 1 is red, 2 is yellow, 3 is green, 4 is blue, and 5 is violet) for various values of .The..
The sequence defined by and the quadratic recurrence equation(1)This sequence arises in Euclid's proof that there are an infinite number of primes. The proof proceeds by constructing a sequence of primes using the recurrence relation(2)(Vardi 1991). Amazingly, there is a constant(3)(OEIS A076393) such that(4)(Aho and Sloane 1973, Vardi 1991, Graham et al. 1994). The first few numbers in Sylvester's sequence are 2, 3, 7, 43, 1807, 3263443, 10650056950807, ... (OEIS A000058). The satisfy(5)In addition, if is an irrational number, then the th term of an infinite sum of unit fractions used to represent as computed using the greedy algorithm must be smaller than .The of the first few prime are 0, 1, 2, 3, 5, ..., corresponding to 2, 3, 7, 43, 3263443, ... (OEIS A014546). Vardi (1991) gives a lists of factors less than of for and shows that is composite for . Furthermore, all numbers less than in Sylvester's sequence are squarefree, and no squareful..
The integer sequence defined by the recurrencerelation(1)with the initial conditions . This is the same recurrence relation as for the Perrin sequence, but with different initial conditions.The recurrence relation can be solved explicitly,giving(2)where is the th root of(3)Another form of the solution is(4)where is the th root of(5)The first few terms are 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, ... (OEIS A000931).The first few prime Padovan numbers are 2, 2, 3, 5, 7, 37, 151, 3329, 23833, ... (OEIS A100891), corresponding to indices ,3, 4, 5, 7, 8, 14, 19, 30, 37, 84, 128, 469, 666, 1262, 1573, 2003, 2210, 2289, 4163, 5553, 6567, 8561, 11230, 18737, 35834, 44259, 536485, ... (OEIS A112882). The search for prime numerators has been completed up to by E. W. Weisstein (Apr. 10, 2011), and the following table summarizes the largest known values.decimal digitsdiscoverer53648565518E. W. Weisstein (May 16, 2009)72773488874E. W. Weisstein..
The Somos sequences are a set of related symmetrical recurrence relations which, surprisingly, always give integers. The Somos sequence of order , or Somos- sequence, is defined by(1)where is the floor function and for , ..., .The 2- and 3-Somos sequences consist entirely of 1s. The -Somos sequences for , 5, 6, and 7 are(2)(3)(4)(5)The first few terms are summarized in the following table. OEIS, , ...4A0067201, 1, 1, 1, 2, 3, 7, 23, 59, 314, 1529, ...5A0067211, 1, 1, 1, 1, 2, 3, 5, 11, 37, 83, 274, 1217, ...6A0067221, 1, 1, 1, 1, 1, 3, 5, 9, 23, 75, 421, 1103, ...7A0067231, 1, 1, 1, 1, 1, 1, 3, 5, 9, 17, 41, 137, 769, ...Combinatorial interpretations for Somos-4 and Somos-5 were found by Speyer (2004) and for Somos-6 and Somos-7 by Carroll and Speyer (2004).Gale (1991) gives simple proofs of the integer-only property of the Somos-4 and Somos-5 sequences, and attributes the first proof to Janice Malouf. In unpublished work, Hickerson and Stanley independently..
Consider the recurrence relation(1)with . The first few iterates of are 1, 2, 3, 5, 10, 28, 154, ... (OEIS A003504). The terms grow extremely rapidly, but are given by the asymptotic formula(2)(OEIS A116603; correcting Finch 2003, p. 446),where(3)(OEIS A115632; Finch 2003, p. 446; Zagier).It is more convenient to work with the transformed sequence(4)which gives the new recurrence(5)with initial condition . Now, will be nonintegral iff . The smallest for which (mod ) therefore gives the smallest nonintegral . In addition, since , is also the smallest nonintegral .For example, we have the sequences :(6)(7)(8)Testing values of shows that the first nonintegral is . Note that a direct verification of this fact is impossible since(9)(calculated using the asymptotic formula) is much too large to be computed and stored explicitly.A sequence even more striking for assuming integer values only for finitely many terms is the 3-Göbel..
A recursive sequence , also known as a recurrence sequence, is a sequence of numbers indexed by an integer and generated by solving a recurrence equation. The terms of a recursive sequences can be denoted symbolically in a number of different notations, such as , , or f, where is a symbol representing the sequence.The idea of sequences in which later terms are deduced from earlier ones, which is implicit in the principle of mathematical induction, dates to antiquity.In the case of linear recurrence equationssuch as the recurrence(with ) generating the Fibonacci numbers, it is possible to solve for an explicit analytic form of the th term of the sequence. Some special classes of recurrence equations have analytic solutions for specific parameters, but solutions for a general parameter is not known. An example of this type is the logistic equationwhich has known exact solutions only for , 2, and 4. It is not known how to solve a general recurrence..
Let a sequence be defined by(1)(2)(3)(4)Also define the associated polynomial(5)and let be its discriminant. The Perrin sequence is a special case corresponding to . The signature mod of an integer with respect to the sequence is then defined as the 6-tuple (, , , , , ) (mod ). 1. An integer has an S-signature if its signature (mod ) is (, , , , , ). 2. An integer has a Q-signature if its signature (mod ) is congruent to () where, for some integer with , , , and . 3. An integer has an I-signature if its signature (mod ) is congruent to (), where and .
For a general second-order linear recurrenceequation(1)define a multiplication rule on ordered pairs by(2)The inverse is then given by(3)and we have the identity(4)(Beeler et al. 1972, Item 12).
A recurrence equation (also called a difference equation) is the discrete analog of a differential equation. A difference equation involves an integer function in a form like(1)where is some integer function. The above equation is the discrete analog of the first-order ordinary differential equation(2)Examples of difference equations often arise in dynamical systems. Examples include the iteration involved in the Mandelbrot and Julia set definitions,(3)with a constant, as well as the logistic equation(4)with a constant. Perhaps the most famous example of a recurrence relation is the one defining the Fibonacci numbers,(5)for and with .Recurrence equations can be solved using RSolve[eqn, a[n], n]. The solutions to a linear recurrence equation can be computed straightforwardly, but quadratic recurrence equations are not so well understood.The sequence generated by a recurrence relation is called a recurrence sequence.Let(6)where..
Given the generating functions defined by(1)(2)(3)(OEIS A051028, A051029,and A051030), then(4)Hirschhorn (1995) showed that(5)(6)(7)where(8)(9)Hirschhorn (1996) showed that checking the first seven cases to 6 is sufficient to prove the result.
A quadratic recurrence is a recurrence equation on a sequence of numbers expressing as a second-degree polynomial in with . For example,is a quadratic recurrence equation.A quadratic recurrence equation of the formin which no cross terms are present is known as a quadraticmap.
A linear recurrence equation is a recurrence equation on a sequence of numbers expressing as a first-degree polynomial in with . For example(1)A quotient-difference table eventually yields a line of 0s iff the starting sequence is defined by a linear recurrence equation.The Wolfram Language command LinearRecurrence[ker, init, n] gives the sequence of length obtained by iterating the linear recurrence with kernel ker starting with initial values init, where for example a kernel denotes the recurrence relation and the initial values are . FindLinearRecurrence[list] attempts to find a minimal linear recurrence that generates list. RecurrenceTable[eqns, expr, n, nmax] generates a list of values of expr for successive based on solving specified the recurrence equations.The following table summarizes some common linear recurrence equations and the corresponding solutions.recurrenceinitial conditionssolutionFibonacci number..
A quadratic map is a quadratic recurrenceequation of the form(1)While some quadratic maps are solvable in closed form (for example, the three solvablecases of the logistic map), most are not.A simple example of a quadratic map with a closed-form solution is(2)with , which has solution , the first few terms of which for , 1, ... are 2, 4, 16, 256, 65536, 4294967296, ... (OEIS A001146).Another example is the number of "strongly" binary trees of height , given by(3)with . The first few terms are 2, 5, 26, 677, 458330, 210066388901, 44127887745906175987802, ... (OEIS A003095) This recurrence has the "analytic" solution(4)where(5)(6)(OEIS A077496) and is the floor function (Aho and Sloane 1973).A third example is the closest strict underapproximation of the number 1,(7)where are integers. The solution is given by the recurrence(8)with . The resulting sequence is known as Sylvester's sequence and has first few terms 2,..
The Catalan numbers on nonnegative integers are a set of numbers that arise in tree enumeration problems of the type, "In how many ways can a regular -gon be divided into triangles if different orientations are counted separately?" (Euler's polygon division problem). The solution is the Catalan number (Pólya 1956; Dörrie 1965; Honsberger 1973; Borwein and Bailey 2003, pp. 21-22), as graphically illustrated above (Dickau).Catalan numbers are commonly denoted (Graham et al. 1994; Stanley 1999b, p. 219; Pemmaraju and Skiena 2003, p. 169; this work) or (Goulden and Jackson 1983, p. 111), and less commonly (van Lint and Wilson 1992, p. 136).Catalan numbers are implemented in the WolframLanguage as CatalanNumber[n].The first few Catalan numbers for , 2, ... are 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, ... (OEIS A000108).Explicit formulas for include(1)(2)(3)(4)(5)(6)(7)where..
The integer sequence defined by the recurrence(1)with the initial conditions , , . This recurrence relation is the same as that for the Padovan sequence but with different initial conditions. The first few terms for , 1, ..., are 3, 0, 2, 3, 2, 5, 5, 7, 10, 12, 17, ... (OEIS A001608).The above cartoon (Amend 2005) shows an unconventional sports application of the Perrin sequence (right panel). (The left two panels instead apply the Fibonacci numbers). is the solution of a third-order linear homogeneous recurrence equation having characteristic equation(2)Denoting the roots of this equation by , , and , with the unique real root, the solution is then(3)Here,(4)is the plastic constant , which is also given by the limit(5)The asymptotic behavior of is(6)The first few primes in this sequence are 2, 3, 2, 5, 5, 7, 17, 29, 277, 367, 853, ... (OEIS A074788), which occur for terms , 3, 4, 5, 6, 7, 10, 12, 20, 21, 24, 34, 38, 75, 122, 166, 236, 355, 356, 930, 1042,..
Apéry's numbers are defined by(1)(2)(3)where is a binomial coefficient. The first few for , 1, 2, ... are 1, 5, 73, 1445, 33001, 819005, ... (OEIS A005259).The first few prime Apéry numbers are 5, 73, 12073365010564729, 10258527782126040976126514552283001, ... (OEIS A092826), which have indices , 2, 12, 24, ... (OEIS A092825).The case of Schmidt's problem expresses these numbers in the form(4)(Strehl 1993, 1994; Koepf 1998, p. 55).They are also given by the recurrence equation(5)with and (Beukers 1987).There is also an associated set of numbers(6)(7)(Beukers 1987), where is a generalized hypergeometric function. The values for , 1, ... are 1, 3, 19, 147, 1251, 11253, 104959, ... (OEIS A005258). The first few prime -numbers are 5, 73, 12073365010564729, 10258527782126040976126514552283001, ... (OEIS A092827), which have indices , 2, 6, 8, ... (OEIS A092828), with no others for (Weisstein, Mar. 8, 2004).The..
The Pell numbers are the numbers obtained by the s in the Lucas sequence with and . They correspond to the Pell polynomial . Similarly, the Pell-Lucas numbers are the s in the Lucas sequence with and , and correspond to the Pell-Lucas polynomial .The Pell numbers and Pell-Lucas numbers are also equal to(1)(2)where is a Fibonacci polynomial.The Pell and Pell-Lucas numbers satisfy the recurrencerelation(3)with initial conditions and for the Pell numbers and for the Pell-Lucas numbers.The th Pell and Pell-Lucas numbers are explicitly given by the Binet-type formulas(4)(5)The th Pell and Pell-Lucas numbers are given by the binomial sums(6)(7)respectively.The Pell and Pell-Lucas numbers satisfy the identities(8)(9)(10)and(11)(12)For , 1, ..., the Pell numbers are 0, 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, ... (OEIS A000129).For a Pell number to be prime, it is necessary that be prime. The indices of (probable) prime Pell numbers are 2, 3, 5,..
A piecewise linear, one-dimensional map on the interval exhibiting chaotic dynamics and given by(1)The first few iterations of (1) give(2)(3)(4)where is the initial value, plotted above through five iterations (with increasing iteration number indicated by colors; 1 is red, 2 is yellow, 3 is green, 4 is blue, and 5 is violet) for various values of .The natural invariant of the tent map is .
Let be a rational number in the closed interval , and generate a sequence using the map(1)Then the number of periodic map orbits of period (for prime) is given by(2)(i.e., the number of period- repeating bit strings, modulo shifts). Since a typical map orbit visits each point with equal probability, the natural invariant is given by(3)