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Centered Decagonal Number
A centered decagonal number is a centered figurate number that represents a decagon with a dot in the center and all other dots surrounding the center dot in successive decagonal layers. The centered decagonal number for ''n'' is given by the formula :5n^2+5n+1 \, Thus, the first few centered decagonal numbers are : 1, 11, 31, 61, 101, 151, 211, 281, 361, 451, 551, 661, 781, 911, 1051, ... Like any other centered ''k''-gonal number, the ''n''th centered decagonal number can be reckoned by multiplying the (''n'' − 1)th triangular number by ''k'', 10 in this case, then adding 1. As a consequence of performing the calculation in base 10, the centered decagonal numbers can be obtained by simply adding a 1 to the right of each triangular number. Therefore, all centered decagonal numbers are odd and in base 10 always end in 1. Another consequence of this relation to triangular numbers is the simple recurrence relation for centered decagonal numbers: :CD ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Centered Decagonal Number
A centered decagonal number is a centered figurate number that represents a decagon with a dot in the center and all other dots surrounding the center dot in successive decagonal layers. The centered decagonal number for ''n'' is given by the formula :5n^2+5n+1 \, Thus, the first few centered decagonal numbers are : 1, 11, 31, 61, 101, 151, 211, 281, 361, 451, 551, 661, 781, 911, 1051, ... Like any other centered ''k''-gonal number, the ''n''th centered decagonal number can be reckoned by multiplying the (''n'' − 1)th triangular number by ''k'', 10 in this case, then adding 1. As a consequence of performing the calculation in base 10, the centered decagonal numbers can be obtained by simply adding a 1 to the right of each triangular number. Therefore, all centered decagonal numbers are odd and in base 10 always end in 1. Another consequence of this relation to triangular numbers is the simple recurrence relation for centered decagonal numbers: :CD ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Centered Number
The centered polygonal numbers are a class of series of figurate numbers, each formed by a central dot, surrounded by polygonal layers of dots with a constant number of sides. Each side of a polygonal layer contains one more dot than each side in the previous layer; so starting from the second polygonal layer, each layer of a centered ''k''-gonal number contains ''k'' more dots than the previous layer. Examples Each centered ''k''-gonal number in the series is ''k'' times the previous triangular number, plus 1. This can be formalized by the expression \frac +1, where ''n'' is the series rank, starting with 0 for the initial 1. For example, each centered square number in the series is four times the previous triangular number, plus 1. This can be formalized by the expression \frac +1. These series consist of the *centered triangular numbers 1, 4, 10, 19, 31, 46, 64, 85, 109, 136, 166, 199, ... (), *centered square numbers 1, 5, 13, 25, 41, 61, 85, 113, 145, 181, 221, 265, ... ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Figurate Number
The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes (polygonal numbers) and different dimensions (polyhedral numbers). The term can mean * polygonal number * a number represented as a discrete -dimensional regular geometry, geometric pattern of -dimensional Ball (mathematics), balls such as a polygonal number (for ) or a polyhedral number (for ). * a member of the subset of the sets above containing only triangular numbers, pyramidal numbers, and their analogs in other dimensions. Terminology Some kinds of figurate number were discussed in the 16th and 17th centuries under the name "figural number". In historical works about Greek mathematics the preferred term used to be ''figured number''. In a use going back to Jacob Bernoulli's Ars Conjectandi, the term ''figurate number'' is used for triangular numbers made up of successive integers, tetrahedral numbers made up of successi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decagon
In geometry, a decagon (from the Greek δέκα ''déka'' and γωνία ''gonía,'' "ten angles") is a ten-sided polygon or 10-gon.. The total sum of the interior angles of a simple decagon is 1440°. A self-intersecting ''regular decagon'' is known as a decagram. Regular decagon A '' regular decagon'' has all sides of equal length and each internal angle will always be equal to 144°. Its Schläfli symbol is and can also be constructed as a truncated pentagon, t, a quasiregular decagon alternating two types of edges. Side length The picture shows a regular decagon with side length a and radius R of the circumscribed circle. * The triangle E_E_1M has to equally long legs with length R and a base with length a * The circle around E_1 with radius a intersects ]M\,E_ _in_a_point_P_(not_designated_in_the_picture)._ *_Now_the_triangle_\;_is_a_isosceles_triangle.html" ;"title="/math> in a point P (not designated in the picture). * Now the triangle \; is a isosceles triang ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1 (number)
1 (one, unit, unity) is a number representing a single or the only entity. 1 is also a numerical digit and represents a single unit of counting or measurement. For example, a line segment of ''unit length'' is a line segment of length 1. In conventions of sign where zero is considered neither positive nor negative, 1 is the first and smallest positive integer. It is also sometimes considered the first of the infinite sequence of natural numbers, followed by 2, although by other definitions 1 is the second natural number, following 0. The fundamental mathematical property of 1 is to be a multiplicative identity, meaning that any number multiplied by 1 equals the same number. Most if not all properties of 1 can be deduced from this. In advanced mathematics, a multiplicative identity is often denoted 1, even if it is not a number. 1 is by convention not considered a prime number; this was not universally accepted until the mid-20th century. Additionally, 1 is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
11 (number)
11 (eleven) is the natural number following 10 and preceding 12. It is the first repdigit. In English, it is the smallest positive integer whose name has three syllables. Name "Eleven" derives from the Old English ', which is first attested in Bede's late 9th-century ''Ecclesiastical History of the English People''. It has cognates in every Germanic language (for example, German ), whose Proto-Germanic ancestor has been reconstructed as , from the prefix (adjectival " one") and suffix , of uncertain meaning. It is sometimes compared with the Lithuanian ', though ' is used as the suffix for all numbers from 11 to 19 (analogously to "-teen"). The Old English form has closer cognates in Old Frisian, Saxon, and Norse, whose ancestor has been reconstructed as . This was formerly thought to be derived from Proto-Germanic (" ten"); it is now sometimes connected with or ("left; remaining"), with the implicit meaning that "one is left" after counting to ten.''Oxford English Dic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
31 (number)
31 (thirty-one) is the natural number following thirty, 30 and preceding 32 (number), 32. It is a prime number. In mathematics 31 is the 11th prime number. It is a superprime and a Self number#Self primes, self prime (after 3, 5, and 7), as no integer added up to its base 10 digits results in 31. It is a lucky prime and a happy number; two properties it shares with 13 (number), 13, which is its dual emirp and permutable prime. 31 is also a primorial prime, like its twin prime, 29 (number), 29. 31 is the number of regular polygons with an odd number of sides that are known to be constructible polygon, constructible with compass and straightedge, from combinations of known Fermat primes of the form 22''n'' + 1. 31 is the third Mersenne prime of the form 2''n'' − 1. It is also the eighth Mersenne prime exponent, specifically for the number 2,147,483,647, which is the maximum positive value for a 32-bit Integer (computer science), signed binary integer in computing. After 3, it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
61 (number)
61 (sixty-one) is the natural number following 60 and preceding 62. In mathematics 61 is: *the 18th prime number. *a twin prime with 59. *a cuban prime of the form ''p'' = , where ''x'' = ''y'' + 1. *the smallest ''proper prime'', a prime ''p'' which ends in the digit 1 in base 10 and whose reciprocal in base 10 has a repeating sequence with length ''p'' − 1. In such primes, each digit 0, 1, ..., 9 appears in the repeating sequence the same number of times as does each other digit (namely, times). *the exponent of the 9th Mersenne prime. (261 − 1 = ) *the sum of two squares, 52 + 62. *a centered square number. *a centered hexagonal number. *a centered decagonal number. *the sixth Euler zigzag number (or Up/down number). *a unique prime in base 14, since no other prime has a 6-digit period in base 14. *a Pillai prime since 8! + 1 is divisible by 61 but 61 is not one more than a multiple of 8. *a Keith number, because it recurs in a Fibonacci-like sequence started from i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
101 (number)
101 (one hundred [and] one) is the natural number following 100 (number), 100 and preceding 102 (number), 102. It is variously pronounced "one hundred and one" / "a hundred and one", "one hundred one" / "a hundred one", and "one oh one". As an Ordinal number (linguistics), ordinal number, 101st (one hundred [and] first), rather than 101th, is the correct form. In mathematics 101 is: *the 26th prime number, and the smallest above 100. *a palindromic number in base 10, and so a palindromic prime. *a Chen prime since 103 (number), 103 is also prime, with which it makes a twin prime pair. *a sexy prime since 107 and 113 are also prime, with which it makes a sexy prime triplet. *a unique prime, because the period length of its reciprocal is unique among primes. *an Eisenstein prime with no imaginary part and real part of the form 3n - 1. *the fifth alternating factorial. *a centered decagonal number. *the only existing prime with alternating 1s and 0s in base 10 and the largest known ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
151 (number)
151 (one hundred ndfifty-one) is a natural number. It follows 150 and precedes 152. In mathematics 151 is the 36th prime number, the previous is 149, with which it comprises a twin prime. 151 is also a palindromic prime, a centered decagonal number, and a lucky number. 151 appears in the Padovan sequence, preceded by the terms 65, 86, 114; it is the sum of the first two of these. 151 is a unique prime in base 2, since it is the only prime with period 15 in base 2. 151 is the number of uniform paracompact honeycombs with infinite facets and vertex figures in the third dimension, which stem from 23 different Coxeter groups. Split into two whole numbers, 151 is the sum of 75 and 76, both of which are also relevant numbers in Euclidean and hyperbolic space: * 75 is the total number of non-prismatic uniform polyhedra, which incorporate regular polyhedra, semiregular polyhedra, and star polyhedra, *75 is also the number of uniform compound polyhedra, inclusive of seve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
911 (number)
911 (nine hundred ndeleven) is the integer following 910 and preceding 912. It is a prime number, a Sophie Germain prime and the sum of three consecutive primes (293 + 307 + 311). It is an Eisenstein prime with no imaginary part and real part of the form 3''n'' − 1. Since 913 is a semiprime, 911 is a Chen prime. It is also a centered decagonal number. There are 911 inverse semigroups of order 7 911 is obtained by concatenating its product of digits and sum of digits. See also * 9-1-1, a North American emergency telephone number * 9/11, the September 11 attacks of 2001 * AD 911 __NOTOC__ 911 ( CMXI) was a common year starting on Tuesday (link will display the full calendar) of the Julian calendar. Events By place Europe * September 24 — King Louis IV (the Child), the last ruler of the Carolingian Dynasty ..., a year * Porsche 911, a sports car References {{DEFAULTSORT:911 (Number) Integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 sequence of triangular numbers, starting with the 0th triangular number, is (This sequence is included in the On-Line Encyclopedia of Integer Sequences .) Formula The triangular numbers are given by the following explicit formulas: T_n= \sum_^n k = 1+2+3+ \dotsb +n = \frac = , where \textstyle is 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 first equation can be illustrated using a visual proof. For every triangular number T_n, imagine a "half-square" arrangement of objects corresponding to the triangular numb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
