|
Padovan Number
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 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 in his Scientific American column ''Mathematical Recreations'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."German original: "Die Mathematik ist die Königin der Wissenschaften, und die Arithmetik ist die Königin der Mathematik." Number theorists study prime numbers as well as the properties of mathematical objects made out of integers (for example, rational numbers) or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory are often best understood through the study of Complex analysis, analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pisot–Vijayaraghavan Number
In mathematics, a Pisot–Vijayaraghavan number, also called simply a Pisot number or a PV number, is a real algebraic integer greater than 1, all of whose Galois conjugates are less than 1 in absolute value. These numbers were discovered by Axel Thue in 1912 and rediscovered by G. H. Hardy in 1919 within the context of diophantine approximation. They became widely known after the publication of Charles Pisot's dissertation in 1938. They also occur in the uniqueness problem for Fourier series. Tirukkannapuram Vijayaraghavan and Raphael Salem continued their study in the 1940s. Salem numbers are a closely related set of numbers. A characteristic property of PV numbers is that their powers approach integers at an exponential rate. Pisot proved a remarkable converse: if ''α'' > 1 is a real number such that the sequence : \, \alpha^n\, measuring the distance from its consecutive powers to the nearest integer is square-summable, or ''ℓ'' 2, then ''α'' is a Pisot n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pascal's Triangle
In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in India, Persia, China, Germany, and Italy. The rows of Pascal's triangle are conventionally enumerated starting with row n = 0 at the top (the 0th row). The entries in each row are numbered from the left beginning with k = 0 and are usually staggered relative to the numbers in the adjacent rows. The triangle may be constructed in the following manner: In row 0 (the topmost row), there is a unique nonzero entry 1. Each entry of each subsequent row is constructed by adding the number above and to the left with the number above and to the right, treating blank entries as 0. For example, the initial number of row 1 (or any other row) is 1 (the sum of 0 and 1), whereas the numbers 1 and 3 in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Erv Wilson
Ervin Wilson (June 11, 1928 – December 8, 2016) was a Mexico, Mexican/United States, American (dual citizen) music theory, music theorist. Early life Ervin Wilson was born in a remote area of northwest Chihuahua (state), Chihuahua, Mexico, where he lived until the age of fifteen. His mother taught him to play the reed organ and to read musical notation. He began to compose at an early age, but immediately discovered that some of the sounds he was hearing mentally could not be reproduced by the conventional intervals of the organ. As a teenager he began to read books on Indian music, developing an interest in concepts of raga. While he was in the Air Force in Japan, a chance meeting with a total stranger introduced him to Harmonic, musical harmonics, which changed the course of his life and work. Influenced by the work of Joseph Yasser, Wilson began to think of the musical scale as a living process—like a crystal or plant. He mentored many composers and instrument builders. Wo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Root Of 2
The square root of 2 (approximately 1.4142) is a positive real number that, when multiplied by itself, equals the number 2. It may be written in mathematics as \sqrt or 2^, and is an algebraic number. Technically, it should be called the principal square root of 2, to distinguish it from the negative number with the same property. Geometrically, the square root of 2 is the length of a diagonal across a square with sides of one unit of length; this follows from the Pythagorean theorem. It was probably the first number known to be irrational. The fraction (≈ 1.4142857) is sometimes used as a good rational approximation with a reasonably small denominator. Sequence in the On-Line Encyclopedia of Integer Sequences consists of the digits in the decimal expansion of the square root of 2, here truncated to 65 decimal places: : History The Babylonian clay tablet YBC 7289 (c. 1800–1600 BC) gives an approximation of in four sexagesimal figures, , which is accurate to about six ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Padovan Cuboid Spiral
In mathematics the Padovan cuboid spiral is the spiral created by joining the diagonals of faces of successive cuboids added to a unit cube. The cuboids are added sequentially so that the resulting cuboid has dimensions that are successive Padovan numbers... See in particular pp. 96–97.. The first cuboid is 1x1x1. The second is formed by adding to this a 1x1x1 cuboid to form a 1x1x2 cuboid. To this is added a 1x1x2 cuboid to form a 1x2x2 cuboid. This pattern continues, forming in succession a 2x2x3 cuboid, a 2x3x4 cuboid etc. Joining the diagonals of the exposed end of each new added cuboid creates a spiral (seen as the black line in the figure). The points on this spiral In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point. Helices Two major definitions of "spiral" in the American Heritage Dictionary are: [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
L-system
An L-system or Lindenmayer system is a parallel rewriting system and a type of formal grammar. An L-system consists of an alphabet of symbols that can be used to make strings, a collection of production rules that expand each symbol into some larger string of symbols, an initial "axiom" string from which to begin construction, and a mechanism for translating the generated strings into geometric structures. L-systems were introduced and developed in 1968 by Aristid Lindenmayer, a Hungarian theoretical biologist and botanist at the University of Utrecht. Lindenmayer used L-systems to describe the behaviour of plant cells and to model the growth processes of plant development. L-systems have also been used to model the morphology of a variety of organisms and can be used to generate self-similar fractals. Origins As a biologist, Lindenmayer worked with yeast and filamentous fungi and studied the growth patterns of various types of bacteria, such as the cyanobacteria '' Anabaena ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Fibonacci Polynomials
In mathematics, the Fibonacci polynomials are a polynomial sequence which can be considered as a generalization of the Fibonacci numbers. The polynomials generated in a similar way from the Lucas numbers are called Lucas polynomials. Definition These Fibonacci polynomials are defined by a recurrence relation:Benjamin & Quinn p. 141 :F_n(x)= \begin 0, & \mbox n = 0\\ 1, & \mbox n = 1\\ x F_(x) + F_(x),& \mbox n \geq 2 \end The Lucas polynomials use the same recurrence with different starting values: :L_n(x) = \begin 2, & \mbox n = 0 \\ x, & \mbox n = 1 \\ x L_(x) + L_(x), & \mbox n \geq 2. \end They can be defined for negative indices bySpringer :F_(x)=(-1)^F_(x), :L_(x)=(-1)^nL_(x). The Fibonacci polynomials form a sequence of orthogonal polynomials with A_n=C_n=1 and B_n=0. Examples The first few Fibonacci polynomials are: :F_0(x)=0 \, :F_1(x)=1 \, :F_2(x)=x \, :F_3(x)=x^2+1 \, :F_4(x)=x^3+2x \, :F_5(x)=x^4+3x^2+1 \, :F_6(x)=x^5+4x^3+3x \, The first few Lucas polynomials ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fibonacci Numbers
In mathematics, the Fibonacci numbers, commonly denoted , form a 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 numbers include co ... [...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] |
Composition (number Theory)
In mathematics, a composition of an integer ''n'' is a way of writing ''n'' as the sum of a sequence of (strictly) positive integers. Two sequences that differ in the order of their terms define different compositions of their sum, while they are considered to define the same partition of that number. Every integer has finitely many distinct compositions. Negative numbers do not have any compositions, but 0 has one composition, the empty sequence. Each positive integer ''n'' has distinct compositions. A weak composition of an integer ''n'' is similar to a composition of ''n'', but allowing terms of the sequence to be zero: it is a way of writing ''n'' as the sum of a sequence of non-negative integers. As a consequence every positive integer admits infinitely many weak compositions (if their length is not bounded). Adding a number of terms 0 to the ''end'' of a weak composition is usually not considered to define a different weak composition; in other words, weak compositions are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
