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Topic: Pentagonal numbers

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	Pentagonal number - Wikipedia, the free encyclopedia
		Pentagonal numbers are important to Euler's theory of partitions, as expressed in his pentagonal number theorem.
		The nth pentagonal number is one third of the 3n-1th triangular number.
		Pentagonal numbers should not be confused with centered pentagonal numbers.
	en.wikipedia.org /wiki/Pentagonal_number (183 words)

	Abramovich, Fujii, and Wilson article
		Therefore the number of dots in the parallelogram is n(n+1), and the number of dots in the triangle is
		Likewise in the case of square numbers, one can visualize that pentagonal numbers are constituted with triangular numbers: in particular, this sketch presents a pentagonal number of rank 5 as the sum of a triangular number of rank 5 and two triangular numbers of rank 4.
		, are triangular; numbers 5151, 501501, 50015001, 5000150001,...
	jwilson.coe.uga.edu /Texts.Folder/AFW/AFWarticle.html (8744 words)

	Polygonal number (Site not responding. Last check: 2007-10-20)
		In mathematics, a polygonal number is a number that can be arranged as a regular polygon.
		Ancient mathematicians discovered that numbers could be arranged in certain ways when they were represented by pebbles or seeds.
		Polygons with higher numbers of sides, such as pentagons and hexagons, can also be represented as arrangements of dots (by convention 1 is the first polygonal number for any number of sides).
	www.sciencedaily.com /encyclopedia/polygonal_number (573 words)

	The Comenius Project
		For example the number 9 are the last single-digit number and are also a perfect square, 3x3.
		Triangular numbers are the natural numbers which can be drawn as dots and arranged in triangular shape: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, etc.
		Of all numbers, 10 were held in greatest reverence by the Pythagoreans; the sum 1 + 2 + 3 + 4 = 10 were named tetraktys, "the holy fourfouldness", representing the four elements: fire, water, air, and earth.
	home.c2i.net /greaker/comenius/9899/figurenumbers/figuratenumbers2.htm (758 words)

	Fascinating Triangular Numbers By Shyam Sunder Gupta
		Numbers such that d(n), the number of divisors of n, is greater than for any smaller n are called highly composite numbers.
		Numbers such that s(n), the sum of aliquot divisors of n, is greater than n are called Abundant numbers.
		Numbers such that s(n), the sum of aliquot divisors of n, is less than n are called Deficient numbers.
	www.shyamsundergupta.com /triangle.htm (1716 words)

	Ancient Greek Mathematics: Geometric Numbers: Triangular, Quadratic, Pentagonal
		Triangular numbers are so called because they can be arranged in a triangular array of dots (elements).
		Square numbers are so called because they can be arranged as a square array of dots.
		The number 666, also known as the Number of the Beast, is a triangular number.
	www.mlahanas.de /Greeks/FigurativeNumber.htm (366 words)

	Figurate numbers
		Figurate numbers are sequences generated by figures made up of evenly spaced dots.
		Figurate numbers include the triangular numbers, square numbers, pentagonal numbers, etc. Sometimes they are called polygonal numbers since they form polygons.
		The sequence of triangular numbers is generated by dots making right triangles of increasing size.
	math.youngzones.org /FigurateNumbers.html (151 words)

	08/11/99 (Site not responding. Last check: 2007-10-20)
		A partition of the number n is a representation of this number as a sum of natural numbers.
		In addition to pentagonal numbers, there are triangular numbers, square numbers, hexagonal numbers, etc. The initial study of these numbers is attributed to the Pythagoreans, as early as 500 BC.
		We are not surprised that partitions satisfy a recursion relation, although the appearance of pentagonal numbers in the relation is a wonder.
	www.rowan.edu /mars/depts/math/HASSEN/NT/Playpart.html (2490 words)

	Numbers
		Two numbers n and m are called an amicable pair if the sum of all positive divisors of n is equal to the sum of all positive divisors of m and both are equal to n + m.
		Pentagonal numbers to pentagons is the same as triangular numbers to triangles and square numbers to squares.
		Definiton: The number n is called a weird number if it is abundant, but it is not the sum of any subset of its proper factors.
	www.tanyakhovanova.com /Numbers/numbers.html (1939 words)

	Numerical Geometry
		Triangular Number is the sum of all the numbers from 1 to n.
		Square Numbers are also associated the polar concepts of Confinement (due to the limitations imposed by structure) and Expansion (in the sense of the Four Directions).
		This Number is the basis of the Unity Holograph which is built upon four nested word clusters that sum to multiples of the Number (= One = Love).
	www.biblewheel.com /GR/GR_Figurate.asp (1557 words)

	More about Palindromic Pentagonals
		Palindromic numbers are numbers which read the same from
		Pentagonal numbers are defined and calculated by this extraordinary intricate and excruciatingly complex formula.
		Believe it or not but some numbers can't be expressed as the sum of 3 pentagonal numbers.
	www.worldofnumbers.com /penta.htm (221 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		An important function in number theory is EMBED Equation , the number of unrestricted partitions of the positive integer n, that is, the number of ways of writing n as a sum of positive integers.
		Table 1: Partitions of a natural number n nPartitions of np(n)11122, 1+1233, 2+1, 1+1+1344, 3+1, 2+2, 2+1+1, 1+1+1+1555, 4+1, 3+2, 3+1+1, 2+2+1, 2+1+1+1, 1+1+1+1+17 While it is simple to determine EMBED Equation for very small numbers n by actually counting all the partitions, this becomes difficult as the numbers grow.
		Thus the sequence of pentagonal numbers 1, 5, 12, 22, 35, 51,...
	www.rowan.edu /math/HASSEN/Papers/Playingwithpartition.doc (1911 words)

	Numbers
		The cone will be covered by ``parallelograms'', the number of seeds on each side of the parallelogram will (always?) be two neighbouring Fibonacci numbers.
		Marbles are dropped through the top and encounter a number of pins before dropping into cells where they are distributed according to the binomial distribution.
		(Rather than using random numbers, throw a bunch of small objects onto the required area and count the numbers of objects inside the area as a fraction of the total in the rectangular frame).
	camel.math.ca /Education/mpsf/node3.html (650 words)

	Knight's Tour Notes, Part Cc: Chronology 1900 to Present
		In references to these publications the problem numbers are cited at the date proposed, the solution (with diagram) will usually be found in a later issue.
		In many cases the tour was stated as a problem and the solution published in a later issue, consequently it is often necesary to see both entries to understand the tour.
		Their number is wrong, as is pointed out in Comments in the same journal issue by B. McKay.
	www.ktn.freeuk.com /cc.htm (7794 words)

	What Is Information?
		However, in most cases, the numbers represent an underlying message that we are trying to convey: “101” could be translated into the number “5”, or perhaps it symbolizes two state transitions – one from 1 to 0, and the next from 0 to 1.
		For the second number, I studiously made sure that the numbers of ones and zeros matched, and that there weren’t any strings of five digits in a row that were the same.
		In fact, this corresponds to human intuition: people who can memorize numbers easily often comment that how “hard” of easy it is to remember a sequence depends on how easy they can store a mnemonic method in their brain that may have little to do with the statistical properties of the number.
	www.patrickkellogg.com /school/papers/infotheory/index.htm (11148 words)

	Earliest Known Uses of Some of the Words of Mathematics (P)
		Among simple even numbers, some are superabundant, others are deficient: these two classes are as two extremes opposed to one another; as for those that occupy the middle position between the two, they are said to be perfect.
		Figurate number and triangular (as a noun) appear in English in 1706 in William Jones, Synopsis palmariorum matheseos: "The Sums of Numbers in a Continued Arithmetic Proportion from Unity are call'd Figurate...
		It is equal to the square of the number of observations divided by twice the sum of the squares of the errors.
	members.aol.com /jeff570/p.html (13956 words)

	Geometry.Net - Philosophers: Pythagoras Of Samos
		According to Aristotle (see Metaphysics 985b-986a), the Pythagoreans, first to develop the science of mathematics, revered number as the first principle of all things, probably due to their discovery that the principles of musical harmony could be explained with mathematics.
		The cult he founded was devoted to the study of numbers, which the Pythagoreans saw as concrete, material objects.
		They studied figurate numbers defining them as triangular numbers pentagonal numbers hexagonal numbers etc., based on the patterns that numbers of regularly spaced dots formed (Boyer 1968, p.
	www.988.com /philosophers/pythagoras_of_samos.php (3107 words)

	Math Magic - Pentagonal Numbers
		Pentagonal Numbers are numbers that create a pentagon.
		pentagonal number can be found by the formula:
		Here are some ways of manipulating pentagonal numbers:
	www.math-magic.com /misc/pentagonal.htm (79 words)

	π: MATH Pages of Jonathan Vos Post (Site not responding. Last check: 2007-10-20)
		a Pentagonal Number, and the smallest number with 18 divisors
		Of interest are such functions as the smallest number whose complexity exceeds a given value, and upper and lower bounds on the ratio of a number to its complexity.
		Although the phase observable, which is roughly speaking conjugate to the number operator, does not exist as a hermitian operator -- this is a longstanding and still popular problem, initiated by Dirac in 1927 -- it exists as a so-called general observable and permits a probability interpretation via a phase distribution on the unit circle.
	www.magicdragon.com /math.html (4600 words)

	Reply from On-Line Encyclopedia
		ID Number: A000326 (Formerly M3818 and N1562) URL: http://www.research.att.com/projects/OEIS?Anum=A000326 Sequence: 0,1,5,12,22,35,51,70,92,117,145,176,210,247,287,330,376,425, 477,532,590,651,715,782,852,925,1001,1080,1162,1247,1335, 1426,1520,1617,1717,1820,1926,2035,2147,2262,2380,2501,2625, 2752,2882,3015,3151 Name: Pentagonal numbers: n(3n-1)/2.
		Comments: The average of the first n (n>0) pentagonal numbers is the n-th triangular number.
		Euler, On the remarkable properties of the pentagonal numbers Formula: G.f.: x(1+2x)/(1-x)^3.
	www.research.att.com /projects/OEIS?Anum=A000326 (318 words)

	Pentagonal numbers
		Pentagonal numbers related books, DVDs, Music at Amazon
		Pentagonal numbers related discount products at Discount Hunter
		January 1, 1999 -- Mathematics: From the Birth of Numbers.
	www.articlesgalore.com /documents/Category:Pentagonal_numbers (457 words)

	MathFiction (Site not responding. Last check: 2007-10-20)
		A number theorist is suffering from frequent and inexplicable suicide attempts, the latest victim of a small epidemic among academia.
		Solzhenitsyn had been a math major until Hitler and Stalin came up with a different career path for him, and TFC is based on his own brief stay in the luxury side of the Gulag, which he claims saved his...
		This book is made up of notes and e-mail messages from a feminist historian interspersed with chapters from a previously unknown novel by Lord Byron which she has discovered while researching his daughter,...
	www.math.cofc.edu /faculty/kasman/MATHFICT/search.php?go=yes&orderby=title (13000 words)

	Table of Figurate Numbers, Sorted, Through 10,000 (Site not responding. Last check: 2007-10-20)
		It is also the first of an infinite number of multidimensional Polytope Numbers such as the Pentatope (4-Simplex or Hypertetrahedron), Hyperoctahedron, 24-cell, Hyperdodecahedron, and Hypericosahedron -- these Polytope numbers are the subject of a recent paper by this author, and will be given their own web page soon.
		Hendecagonal Number, or 11-gonal number, Hen(16), hence the Heterogeneous Jonathan Number Hen(S(4))
		Hendecagonal Number, or 11-gonal number, Hen(25), hence the Heterogeneous Jonathan Number Hen(S(5))
	www.magicdragon.com /fig.html (10985 words)

	Pentagonal number - Art History Online Reference and Guide
		Pentagonal number - Art History Online Reference and Guide
		Pentagonal number - Your Art History Reference Guide!
		The pentagonal number for n is given by the formula n(3n - 1)/2, with n > 0.
	www.arthistoryclub.com /art_history/Pentagonal_number (213 words)

	MathLinks Math Forum :: View topic - Pentagonal numbers
		Posted: Tue 27 Apr 2004, 18:55 Post subject: Pentagonal numbers
		Challenge: Find a general formula for the nth pentagonal number.
		We see that the changes between consecutive numbers are 4, 7, 10, 13, 16..., which clearly all differ by 3.
	www.mathlinks.ro /Forum/post-83719.html (296 words)

	[No title]
		# The numbers Mxxxx refer to The Encyclopedia of Integer Sequences [SlPl95].
		seq(count(M[1108],size=n),n=0..24); 1, 2, 4, 8, 15, 29, 56, 108, 208, 401, 773, 1490, 2872, 5536, 10671, 20569, 39648, 76424, 147312, 283953, 547337, 1055026, 2033628, 3919944, 7555935 # Pentanacci numbers [PF] M[1122]:=[L,{L=Prod(X,Sequence(Prod(Z,X))),X=Sequence(Z,card
		seq(count(M[1122],size=n),n=0..24); 1, 2, 4, 8, 16, 31, 61, 120, 236, 464, 912, 1793, 3525, 6930, 13624, 26784, 52656, 103519, 203513, 400096, 786568, 1546352, 3040048, 5976577, 11749641 # Hexanacci numbers [PF] M[1128]:=[L,{L=Prod(X,Sequence(Prod(Z,X))),X=Sequence(Z,card
	www.loria.fr /~zimmerma/gaia/jolis (234 words)

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