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Hexagonal Number
A hexagonal number is a figurate number. The ''n''th hexagonal number ''h''''n'' is the number of ''distinct'' dots in a pattern of dots consisting of the ''outlines'' of regular hexagons with sides up to n dots, when the hexagons are overlaid so that they share one vertex (geometry), vertex. The formula for the ''n''th hexagonal number :h_n= 2n^2-n = n(2n-1) = \frac. The first few hexagonal numbers are: :1 (number), 1, 6 (number), 6, 15 (number), 15, 28 (number), 28, 45 (number), 45, 66 (number), 66, 91 (number), 91, 120 (number), 120, 153 (number), 153, 190 (number), 190, 231 (number), 231, 276 (number), 276, 325 (number), 325, 378, 435, 496 (number), 496, 561 (number), 561, 630, 703, 780, 861, 946... Every hexagonal number is a triangular number, but only every ''other'' triangular number (the 1st, 3rd, 5th, 7th, etc.) is a hexagonal number. Like a triangular number, the digital root in base 10 of a hexagonal number can only be 1, 3, 6, or 9. The digital root pattern, repe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Centered Hexagonal Number
In mathematics and combinatorics, a centered hexagonal number, or centered hexagon number, is a centered polygonal number, centered figurate number that represents a hexagon with a dot in the center and all other dots surrounding the center dot in a hexagonal lattice. The following figures illustrate this arrangement for the first four centered hexagonal numbers: : Centered hexagonal numbers should not be confused with hexagonal number, cornered hexagonal numbers, which are figurate numbers in which the associated hexagons share a vertex. The sequence of hexagonal numbers starts out as follows : :1, 7, 19 (number), 19, 37 (number), 37, 61 (number), 61, 91 (number), 91, 127 (number), 127, 169 (number), 169, 217 (number), 217, 271 (number), 271, 331 (number), 331, 397 (number), 397, 469, 547, 631, 721, 817, 919. Formula The th centered hexagonal number is given by the formula :H(n) = n^3 - (n-1)^3 = 3n(n-1)+1 = 3n^2 - 3n +1. \, Expressing the formula as :H(n) = 1+6\left(\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Hexagonal Numbers
A hexagonal number is a figurate number. The ''n''th hexagonal number ''h''''n'' is the number of ''distinct'' dots in a pattern of dots consisting of the ''outlines'' of regular hexagons with sides up to n dots, when the hexagons are overlaid so that they share one vertex. The formula for the ''n''th hexagonal number :h_n= 2n^2-n = n(2n-1) = \frac. The first few hexagonal numbers are: : 1, 6, 15, 28, 45, 66, 91, 120, 153, 190, 231, 276, 325, 378, 435, 496, 561, 630, 703, 780, 861, 946... Every hexagonal number is a triangular number, but only every ''other'' triangular number (the 1st, 3rd, 5th, 7th, etc.) is a hexagonal number. Like a triangular number, the digital root in base 10 of a hexagonal number can only be 1, 3, 6, or 9. The digital root pattern, repeating every nine terms, is "1 6 6 1 9 3 1 3 9". Every even perfect number is hexagonal, given by the formula :M_p 2^ = M_p \frac = h_=h_ :where ''M''''p'' is a Mersenne prime. No odd perfect numbers are known ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Triangular Number
A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots in the triangular arrangement with dots on each side, and is equal to the sum of the natural numbers from 1 to . The first 100 terms sequence of triangular numbers, starting with the 0th triangular number, are Formula The triangular numbers are given by the following explicit formulas: where \textstyle is notation for a binomial coefficient. It represents the number of distinct pairs that can be selected from objects, and it is read aloud as " plus one choose two". The fact that the nth triangular number equals n(n+1)/2 can be illustrated using a visual proof. For every triangular number T_n, imagine a "half-rectangle" arrangement of objects corresponding to the triangular number, as in the figure below. Copying this arrangement ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Hexadecimal
Hexadecimal (also known as base-16 or simply hex) is a Numeral system#Positional systems in detail, positional numeral system that represents numbers using a radix (base) of sixteen. Unlike the decimal system representing numbers using ten symbols, hexadecimal uses sixteen distinct symbols, most often the symbols "0"–"9" to represent values 0 to 9 and "A"–"F" to represent values from ten to fifteen. Software developers and system designers widely use hexadecimal numbers because they provide a convenient representation of binary code, binary-coded values. Each hexadecimal digit represents four bits (binary digits), also known as a nibble (or nybble). For example, an 8-bit byte is two hexadecimal digits and its value can be written as to in hexadecimal. In mathematics, a subscript is typically used to specify the base. For example, the decimal value would be expressed in hexadecimal as . In programming, several notations denote hexadecimal numbers, usually involving a prefi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
325 (number)
325 (three hundred ndtwenty-five) is the natural number following 324 __NOTOC__ Year 324 ( CCCXXIV) was a leap year starting on Wednesday in the Julian calendar. At the time, it was known as the Year of the Consulship of Crispus and Constantinus (or, less frequently, year 1077 ''Ab urbe condita''). The denominati ... and preceding 326. In mathematics * 325 is the smallest number to be the sum of two squares in 3 different ways: 12 + 182, 62 + 172 and 102 + 152. * 325 is the smallest (and only known) 3- hyperperfect number. References Integers {{Num-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Empty Sum
In mathematics, an empty sum, or nullary sum, is a summation where the number of terms is zero. The natural way to extend non-empty sums is to let the empty sum be the additive identity. Let a_1, a_2, a_3, ... be a sequence of numbers, and let :s_m = \sum_^m a_i = a_1 + \cdots + a_m be the sum of the first ''m'' terms of the sequence. This satisfies the recurrence :s_m = s_ + a_m provided that we use the following natural convention: s_0=0. In other words, a "sum" s_1 with only one term evaluates to that one term, while a "sum" s_0 with no terms evaluates to 0. Allowing a "sum" with only 1 or 0 terms reduces the number of cases to be considered in many mathematical formulas. Such "sums" are natural starting points in induction proofs, as well as in algorithms. For these reasons, the "empty sum is zero" extension is standard practice in mathematics and computer programming (assuming the domain has a zero element). For the same reason, the empty product is taken to be the multip ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Sigma Notation
In mathematics, summation is the addition of a sequence of numbers, called ''addends'' or ''summands''; the result is their ''sum'' or ''total''. Beside numbers, other types of values can be summed as well: functions, vectors, matrices, polynomials and, in general, elements of any type of mathematical objects on which an operation denoted "+" is defined. Summations of infinite sequences are called series. They involve the concept of limit, and are not considered in this article. The summation of an explicit sequence is denoted as a succession of additions. For example, summation of is denoted , and results in 9, that is, . Because addition is associative and commutative, there is no need for parentheses, and the result is the same irrespective of the order of the summands. Summation of a sequence of only one summand results in the summand itself. Summation of an empty sequence (a sequence with no elements), by convention, results in 0. Very often, the elements of a sequence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Vienna Sausage
Vienna sausage (; Viennese/Austrian German: or ; Swiss German: ; Swabian: or ) is a thin parboiled sausage traditionally made of pork and beef in a casing of sheep's intestine, then given a low-temperature smoking. The word is German for ' Viennese'. In Austria, the term is uncommon for this food item, which instead is usually called . Europe In some European countries, cooked and often smoked wiener sausages bought fresh from supermarkets, delicatessens and butcher shops may be called by a name (such as in German or French) which translates in English as "Vienna sausage". Traditionally, they are made from cured pork, but in Eastern and Southern Europe, sausages made from chicken or turkey are more common; these are also sold in places with a significant population of people who do not eat pork for religious reasons. Wieners sold in Europe have a taste and texture very much like North American hot dogs, but are usually longer and somewhat thinner, with a very light, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative integers. The set (mathematics), set of all integers is often denoted by the boldface or blackboard bold The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the set of natural numbers, the set of integers \mathbb is Countable set, countably infinite. An integer may be regarded as a real number that can be written without a fraction, fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , 5/4, and Square root of 2, are not. The integers form the smallest Group (mathematics), group and the smallest ring (mathematics), ring containing the natural numbers. In algebraic number theory, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Adrien-Marie Legendre
Adrien-Marie Legendre (; ; 18 September 1752 – 9 January 1833) was a French people, French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as the Legendre polynomials and Legendre transformation are named after him. He is also known for his contributions to the Least squares, method of least squares, and was the first to officially publish on it, though Carl Friedrich Gauss had discovered it before him. Life Adrien-Marie Legendre was born in Paris on 18 September 1752 to a wealthy family. He received his education at the Collège Mazarin in Paris, and defended his thesis in physics and mathematics in 1770. He taught at the École Militaire in Paris from 1775 to 1780 and at the École Normale Supérieure, École Normale from 1795. At the same time, he was associated with the Bureau des Longitudes. In 1782, the Prussian Academy of Sciences, Berlin Academy awarded Legendre a prize for his treatise on projectiles in resistant m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
130 (number)
130 (one hundred ndthirty) is the natural number following 129 and preceding 131. In mathematics 130 is a sphenic number. It is a noncototient In number theory, a noncototient is a positive integer that cannot be expressed as the difference between a positive integer and the number of coprime integers below it. That is, , where stands for Euler's totient function In number theory ... since there is no answer to the equation ''x'' - φ(''x'') = 130. 130 is the only integer that is the sum of the squares of its first four divisors, including 1: 12 + 22 + 52 + 102 = 130. 130 is the largest number that cannot be written as the sum of four hexagonal numbers. 130 equals both 27 + 2 and 53 + 5 and is therefore a ''doubly strictly '' number. In other fields * A 130-30 fund or a ratio up to 150/50 is a type of collective investment vehicle References {{DEFAULTSORT:130 (Number) Integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Mersenne Prime
In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century. If is a composite number then so is . Therefore, an equivalent definition of the Mersenne primes is that they are the prime numbers of the form for some prime . The exponents which give Mersenne primes are 2, 3, 5, 7, 13, 17, 19, 31, ... and the resulting Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, ... . Numbers of the form without the primality requirement may be called Mersenne numbers. Sometimes, however, Mersenne numbers are defined to have the additional requirement that should be prime. The smallest composite Mersenne number with prime exponent ''n'' is . Mersenne primes were studied in antiquity because of their close connection to perfect numbers: the Euclid–Eule ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
