|
Pentagonal Number
A pentagonal number is a figurate number that extends the concept of triangular number, triangular and square numbers to the pentagon, but, unlike the first two, the patterns involved in the construction of pentagonal numbers are not rotational symmetry, rotationally symmetrical. The ''n''th pentagonal number ''pn'' is the number of ''distinct'' dots in a pattern of dots consisting of the ''outlines'' of regular pentagons with sides up to n dots, when the pentagons are overlaid so that they share one vertex (geometry), vertex. For instance, the third one is formed from outlines comprising 1, 5 and 10 dots, but the 1, and 3 of the 5, coincide with 3 of the 10 – leaving 12 distinct dots, 10 in the form of a pentagon, and 2 inside. ''p''n is given by the formula: :p_n = =\binom+3\binom for ''n'' ≥ 1. The first few pentagonal numbers are: 1 (number), 1, 5 (number), 5, 12 (number), 12, 22 (number), 22, 35 (number), 35, 51 (number), 51, 70 (number), 70, 92 (number), 92, 117 (nu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
176 (number)
176 (one hundred ndseventy-six) is the natural number following 175 and preceding 177. In mathematics 176 is an even number and an abundant number. It is an odious number, a self number, a semiperfect number, and a practical number. 176 is a cake number, a happy number, a pentagonal number, and an octagonal number In mathematics, an octagonal number is a figurate number. The ''n''th octagonal number ''o'n'' is the number of dots in a pattern of dots consisting of the outlines of regular octagons with sides up to ''n'' dots, when the octagons are overlai .... 15 can be partitioned in 176 ways. The Higman–Sims group can be constructed as a doubly transitive permutation group acting on a geometry containing 176 points, and it is also the symmetry group of the largest possible set of equiangular lines in 22 dimensions, which contains 176 lines. References External links Number Facts and Trivia: 176The Number 176The Positive Integer 176 Number Gossip: 176 {{D ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive integers Some authors acknowledge both definitions whenever convenient. Sometimes, the whole numbers are the natural numbers as well as zero. In other cases, the ''whole numbers'' refer to all of the integers, including negative integers. The counting numbers are another term for the natural numbers, particularly in primary education, and are ambiguous as well although typically start at 1. The natural numbers are used for counting things, like "there are ''six'' coins on the table", in which case they are called ''cardinal numbers''. They are also used to put things in order, like "this is the ''third'' largest city in the country", which are called ''ordinal numbers''. Natural numbers are also used as labels, like Number (sports), jersey ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sums Of Reciprocals
In mathematics and especially number theory, the sum of reciprocals (or sum of inverses) generally is computed for the reciprocals of some or all of the positive integers (counting numbers)—that is, it is generally the sum of unit fractions. If infinitely many numbers have their reciprocals summed, generally the terms are given in a certain sequence and the first ''n'' of them are summed, then one more is included to give the sum of the first ''n''+1 of them, etc. If only finitely many numbers are included, the key issue is usually to find a simple expression for the value of the sum, or to require the sum to be less than a certain value, or to determine whether the sum is ever an integer. For an infinite series of reciprocals, the issues are twofold: First, does the sequence of sums diverge—that is, does it eventually exceed any given number—or does it converge, meaning there is some number that it gets arbitrarily close to without ever exceeding it? (A set of posit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentagonal Number Visual Proof
In geometry, a pentagon () is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simple or self-intersecting. A self-intersecting ''regular pentagon'' (or ''star pentagon'') is called a pentagram. Regular pentagons A '' regular pentagon'' has Schläfli symbol and interior angles of 108°. A '' regular pentagon'' has five lines of reflectional symmetry, and rotational symmetry of order 5 (through 72°, 144°, 216° and 288°). The diagonals of a convex regular pentagon are in the golden ratio to its sides. Given its side length t, its height H (distance from one side to the opposite vertex), width W (distance between two farthest separated points, which equals the diagonal length D) and circumradius R are given by: :\begin H &= \frac~t \approx 1.539~t, \\ W= D &= \frac~t\approx 1.618~t, \\ W &= \sqrt \cdot H\approx 1.051~H, \\ R &= \sqrt t\approx 0.8507~t, \\ D &= R\ = 2R\cos 18^\cir ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Composition (combinatorics)
In mathematics, a composition of an integer ''n'' is a way of writing ''n'' as the summation, 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 integer 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 2''n''−1 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentagonal Number Theorem
In mathematics, Euler's pentagonal number theorem relates the product and series representations of the Euler function. It states that :\prod_^\left(1-x^\right)=\sum_^\left(-1\right)^x^=1+\sum_^\infty(-1)^k\left(x^+x^\right). In other words, :(1-x)(1-x^2)(1-x^3) \cdots = 1 - x - x^2 + x^5 + x^7 - x^ - x^ + x^ + x^ - \cdots. The exponents 1, 2, 5, 7, 12, ... on the right hand side are given by the formula for ''k'' = 1, −1, 2, −2, 3, ... and are called (generalized) pentagonal numbers . (The constant term 1 corresponds to k=0.) This holds as an identity of convergent power series for , x, ''s'', take the rightmost 45-degree line and move it to form a new row, as in the matching diagram below. : If m ≤ s (as in our newly formed diagram where ''m'' = 2, ''s'' = 5) we may reverse the process by moving the bottom row to form a new 45 degree line (adding 1 element to each of the first ''m'' rows), taking us back to the first diagram. A bit of thought shows that this ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer Partition
In number theory and combinatorics, a partition of a non-negative integer , also called an integer partition, is a way of writing as a summation, sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition. (If order matters, the sum becomes a composition (combinatorics), composition.) For example, can be partitioned in five distinct ways: : : : : : The only partition of zero is the empty sum, having no parts. The order-dependent composition is the same partition as , and the two distinct compositions and represent the same partition as . An individual summand in a partition is called a part. The number of partitions of is given by the Partition function (number theory), partition function . So . The notation means that is a partition of . Partitions can be graphically visualized with Young diagrams or Ferrers diagrams. They occur in a number of branches of mathematics and physics, including the study of symm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler
Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential discoveries in many other branches of mathematics, such as analytic number theory, complex analysis, and infinitesimal calculus. He also introduced much of modern mathematical terminology and notation, including the notion of a mathematical function. He is known for his work in mechanics, fluid dynamics, optics, astronomy, and music theory. Euler has been called a "universal genius" who "was fully equipped with almost unlimited powers of imagination, intellectual gifts and extraordinary memory". He spent most of his adult life in Saint Petersburg, Russia, and in Berlin, then the capital of Prussia. Euler is credited for popularizing the Greek letter \pi (lowercase pi) to denote the ratio of a circle's circumference to its diameter, as we ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
