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Padovan Polynomials
In mathematics, Padovan polynomials are a generalization of Padovan sequence numbers. These polynomials are defined by: :P_n(x) = \begin 1, &\mboxn=1\\ 0, &\mboxn=2\\ x, &\mboxn=3\\ xP_(x)+P_(x),&\mbox n\ge4. \end The first few Padovan polynomials are: :P_1(x)=1 \, :P_2(x)=0 \, :P_3(x)=x \, :P_4(x)=1 \, :P_5(x)=x^2 \, :P_6(x)=2x \, :P_7(x)=x^3+1 \, :P_8(x)=3x^2 \, :P_9(x)=x^4+3x \, :P_(x)=4x^3+1\, :P_(x)=x^5+6x^2.\, The Padovan numbers are recovered by evaluating the polynomials P''n''−3(''x'') at ''x'' = 1. Evaluating P''n''−3(''x'') at ''x'' = 2 gives the ''n''th Fibonacci number plus (−1)''n''. The ordinary generating function for the sequence is : \sum_^\infty P_n(x) t^n = \frac{1 - x t^2 - t^3} . See also *Polynomial sequence In mathematics, a polynomial sequence is a sequence of polynomials indexed by the nonnegative integers 0, 1, 2, 3, ..., in which each index is equal to the de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Padovan Sequence
In number theory, the Padovan sequence is the sequence of integers ''P''(''n'') defined. by the initial values :P(0)=P(1)=P(2)=1, and the recurrence relation :P(n)=P(n-2)+P(n-3). The first few values of ''P''(''n'') are :1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265, ... A Padovan prime is a Padovan number that's also prime. The first Padovan primes are: :2, 3, 5, 7, 37, 151, 3329, 23833, 13091204281, 3093215881333057, 1363005552434666078217421284621279933627102780881053358473, 1558877695141608507751098941899265975115403618621811951868598809164180630185566719, ... . The Padovan sequence is named after Richard Padovan who attributed its discovery to Netherlands, Dutch architect Hans van der Laan in his 1994 essay ''Dom. Hans van der Laan : Modern Primitive''.Richard Padovan. ''Dom Hans van der Laan: modern primitive'': Architectura & Natura Press, . The sequence was described by Ian Stewart (mathematician), Ian Stewart in his Scientific Ame ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example of a polynomial of a single indeterminate is . An example with three indeterminates is . Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which are central concepts in algebra and algebraic geometry. Etymology The word ''polynomial'' join ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fibonacci Sequence
In mathematics, the Fibonacci numbers, commonly denoted , form a integer sequence, sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from 1 and 1 or sometimes (as did Fibonacci) from 1 and 2. Starting from 0 and 1, the first few values in the sequence are: :0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144. The Fibonacci numbers were first described in Indian mathematics, as early as 200 BC in work by Pingala on enumerating possible patterns of Sanskrit poetry formed from syllables of two lengths. They are named after the Italian mathematician Leonardo of Pisa, later known as Fibonacci, who introduced the sequence to Western European mathematics in his 1202 book ''Liber Abaci''. Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the ''Fibonacci Quarterly''. Applications of Fibonacci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generating Function
In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary series, the ''formal'' power series is not required to converge: in fact, the generating function is not actually regarded as a function, and the "variable" remains an indeterminate. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general linear recurrence problem. One can generalize to formal power series in more than one indeterminate, to encode information about infinite multi-dimensional arrays of numbers. There are various types of generating functions, including ordinary generating functions, exponential generating functions, Lambert series, Bell series, and Dirichlet series; definitions and examples are given below. Every sequence in principle has a generating function of each type (except ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial Sequence
In mathematics, a polynomial sequence is a sequence of polynomials indexed by the nonnegative integers 0, 1, 2, 3, ..., in which each index is equal to the degree of the corresponding polynomial. Polynomial sequences are a topic of interest in enumerative combinatorics and algebraic combinatorics, as well as applied mathematics. Examples Some polynomial sequences arise in physics and approximation theory as the solutions of certain ordinary differential equations: * Laguerre polynomials * Chebyshev polynomials * Legendre polynomials * Jacobi polynomials Others come from statistics: * Hermite polynomials Many are studied in algebra and combinatorics: * Monomials * Rising factorials * Falling factorials * All-one polynomials * Abel polynomials * Bell polynomials * Bernoulli polynomials * Cyclotomic polynomials * Dickson polynomials * Fibonacci polynomials * Lagrange polynomials * Lucas polynomials * Spread polynomials * Touchard polynomials * Rook polynomials Classes of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
