A polynomial given by(1)where are the roots of unity in given by(2)and runs over integers relatively prime to . The prime may be dropped if the product is instead taken over primitive roots of unity, so that(3)The notation is also frequently encountered. Dickson et al. (1923) and Apostol (1975) give extensive bibliographies for cyclotomic polynomials.The cyclotomic polynomial for can also be defined as(4)where is the Möbius function and the product is taken over the divisors of (Vardi 1991, p. 225). is an integer polynomial and an irreducible polynomial with polynomial degree , where is the totient function. Cyclotomic polynomials are returned by the Wolfram Language command Cyclotomic[n, x]. The roots of cyclotomic polynomials lie on the unit circle in the complex plane, as illustrated above for the first few cyclotomic polynomials.The first few cyclotomic polynomials are(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)The cyclotomic..
While the Catalan numbers are the number of p-good paths from to (0,0) which do not cross the diagonal line, the super Catalan numbers count the number of lattice paths with diagonal steps from to (0,0) which do not touch the diagonal line .The super Catalan numbers are given by the recurrencerelation(1)(Comtet 1974), with . (Note that the expression in Vardi (1991, p. 198) contains two errors.) A closed form expression in terms of Legendre polynomials for is(2)(3)(Vardi 1991, p. 199). The first few super Catalan numbers are 1, 1, 3, 11, 45, 197, ... (OEIS A001003). These are often called the "little" Schröder numbers. Multiplying by 2 gives the usual ("large") Schröder numbers 2, 6, 22, 90, ... (OEIS A006318).The first few prime super Catalan numbers have indices 3, 4, 6, 10, 216, ... (OEIS A092839), with no others less than (Weisstein, Mar. 7, 2004), corresponding to the numbers 3, 11, 197,..
The Motzkin numbers enumerate various combinatorial objects. Donaghey and Shapiro (1977) give 14 different manifestations of these numbers. In particular, they give the number of paths from (0, 0) to (, 0) which never dip below and are made up only of the steps (1, 0), (1, 1), and (1, ), i.e., , , and .The first are 1, 2, 4, 9, 21, 51, ... (OEIS A001006). The numbers of decimal digits in for , 1, ... are 1, 4, 45, 473, 4766, 47705, 477113, ... (OEIS A114473), where the digits approach those of (OEIS A114490).The first few prime Motzkin numbers are 2, 127, 15511, 953467954114363, ... (OEIS A092832), which correspond to indices 2, 7, 12, 36, ... (OEIS A092831), with no others for (Weisstein, Mar. 29, 2005).The Motzkin number generating function satisfies(1)and is given by(2)(3)(4) therefore is given by the continued fraction(5)(M. Somos, pers. comm., Apr. 15, 2006).They are given by the recurrence relation(6)with , as well as the nested..
Define the sequence , , and(1)for . The first few values are(2)(3)(4)(5)Janssen and Tjaden (1987) showed that this sequence converges for exactly one value , where (OEIS A085835), confirming Grossman's conjecture. However, no analytic form is known for this constant, either as the root of a function or as a combination of other constants. The plot above shows the first few iterations of for to 30, with odd shown in red and even shown in blue, for ranging from 0 to 1. As can be seen, the solutions alternate by parity. For each fixed , the red values go to 0, while the blue values go to some positive number.Nyerges (2000) has generalized the recurrence to the functional equation(6)
The Schröder number is the number of lattice paths in the Cartesian plane that start at (0, 0), end at , contain no points above the line , and are composed only of steps (0, 1), (1, 0), and (1, 1), i.e., , , and . The diagrams illustrating the paths generating , , and are illustrated above.The numbers are given by the recurrence relation(1)where , and the first few are 2, 6, 22, 90, ... (OEIS A006318). The numbers of decimal digits in for , 1, ... are 1, 7, 74, 761, 7650, 76548, 765543, 7655504, ... (OEIS A114472), where the digits approach those of (OEIS A114491).They have the generating function(2)which satisfies(3)and has closed-form solutions(4)(5)(6)where is a hypergeometric function, is a Gegenbauer polynomial, is a Legendre polynomial, and (5) holds for .The Schröder numbers bear the same relation to the Delannoy numbers as the Catalan numbers do to the binomial coefficients...
A Lucas polynomial sequence is a pair of generalized polynomials which generalize the Lucas sequence to polynomials is given by(1)(2)where(3)(4)Solving for and and taking the solution for with the sign gives(5)(Horadam 1996). Setting gives(6)(7)giving(8)(9)The sequences most commonly considered have , giving(10)The polynomials satisfy the recurrence relation(11)Special cases of the and polynomials are given in the following table.1Fibonacci polynomial Lucas polynomial 1Pell polynomial Pell-Lucas polynomial 1Jacobsthal polynomial Jacobsthal-Lucas polynomial Fermat polynomial Fermat-Lucas polynomial Chebyshev polynomial of the second kind Chebyshev polynomial of the first kind
The Lucas numbers are the sequence of integers defined by the linear recurrence equation(1)with and . The th Lucas number is implemented in the Wolfram Language as LucasL[n].The values of for , 2, ... are 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, ... (OEIS A000204).The Lucas numbers are also a Lucas sequence and are the companions to the Fibonacci numbers and satisfy the same recurrence.The number of ways of picking a set (including the empty set) from the numbers 1, 2, ..., without picking two consecutive numbers (where 1 and are now consecutive) is (Honsberger 1985, p. 122).The only square numbers in the Lucas sequence are 1 and 4 (Alfred 1964, Cohn 1964). The only triangular Lucas numbers are 1, 3, and 5778 (Ming 1991). The only cubic Lucas number is 1.Rather amazingly, if is prime, . The converse does not necessarily hold true, however, and composite numbers such that are known as Lucas pseudoprimes.For , 2, ..., the numbers of decimal digits in are..
The Fibonacci numbers are the sequence of numbers defined by the linear recurrence equation(1)with . As a result of the definition (1), it is conventional to define .The Fibonacci numbers for , 2, ... are 1, 1, 2, 3, 5, 8, 13, 21, ... (OEIS A000045).Fibonacci numbers can be viewed as a particular case of the Fibonacci polynomials with .Fibonacci numbers are implemented in the WolframLanguage as Fibonacci[n].The Fibonacci numbers are also a Lucas sequence , and are companions to the Lucas numbers (which satisfy the same recurrence equation).The above cartoon (Amend 2005) shows an unconventional sports application of the Fibonacci numbers (left two panels). (The right panel instead applies the Perrin sequence).A scrambled version 13, 3, 2, 21, 1, 1, 8, 5 (OEIS A117540) of the first eight Fibonacci numbers appear as one of the clues left by murdered museum curator Jacque Saunière in D. Brown's novel The Da Vinci Code (Brown 2003, pp. 43,..
The Delannoy numbers are the number of lattice paths from to in which only east (1, 0), north (0, 1), and northeast (1, 1) steps are allowed (i.e., , , and ). They are given by the recurrence relation(1)with . The are also given by the sums(2)(3)(4)where is a hypergeometric function.A table for values for the Delannoy numbers is given by(5)(OEIS A008288) for , 1, ... increasing from left to right and , 1, ... increasing from top to bottom.They have the generating function(6)(Comtet 1974, p. 81).Taking gives the central Delannoy numbers , which are the number of "king walks" from the corner of an square to the upper right corner . These are given by(7)where is a Legendre polynomial (Moser 1955; Comtet 1974, p. 81; Vardi 1991). Another expression is(8)(9)(10)where is a binomial coefficient and is a hypergeometric function. These numbers have a surprising connection with the Cantor set (E. W. Weisstein, Apr. 9,..
For a simple continued fraction with convergents , the fundamental recurrence relation is given by
The recurrence relationvalid for , 5, ... with and and which solves the married couples problem (Dörrie 1965, p. 33).
An indicial equation, also called a characteristic equation, is a recurrence equation obtained during application of the Frobenius method of solving a second-order ordinary differential equation. The indicial equation is obtained by noting that, by definition, the lowest order term (that corresponding to ) must have a coefficient of zero. 1. If the two roots are equal, only one solution can beobtained. 2. If the two roots differ by a noninteger, two solutionscan be obtained. 3. If the two roots differ by an integer,the larger will yield a solution. The smaller may or may not. For an example of the construction of an indicial equation, see Besselfunction of the first kind.The following table gives the indicial equations for some common differential equations.differential equationindicial equationBessel differential equationChebyshev differential equationHermite differential equationJacobi differential equationLaguerre differential..
A generalization of the Fibonacci numbers defined by the four constants and the definitions and together with the linear recurrence equationfor . With , , , and , the Horadam sequence reduces to the Fibonacci numbers.