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| 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
| | "Why does my computer keep freezing up?" I was asked by a lot of individuals the cause of their computer freeze difficulties. And I am fed up with spending much time in answering the query time plus time again. This post is to tell you the real cause of your PC Freezes.<br><br>However registry is conveniently corrupted and damaged when you're utilizing your computer. Overtime, without right repair, it usually be loaded with mistakes and incorrect or missing information that usually make a program unable to work properly or apply a certain task. And whenever a system cannot discover the correct information, it usually not learn what to do. Then it freezes up! That is the real cause of your trouble.<br><br>If you compare registry cleaners we want a quickly acting registry cleaning. It's no good spending hours plus the PC waiting for the registry cleaning to complete its task. We want your cleaner to complete its task inside minutes.<br><br>Always see with it that you have installed antivirus, anti-spyware plus anti-adware programs plus have them updated on a regular basis. This can help stop windows XP running slow.<br><br>So to fix this, you really have to be able to make all registry files non-corrupted again. This will dramatically accelerate the loading time of the computer and might allow you to do a large amount of aspects on it again. And fixing these files couldn't be simpler - you only require to utilize a tool called a [http://bestregistrycleanerfix.com/tune-up-utilities tuneup utilities].<br><br>The initially thing you should do is to reinstall any system which shows the error. It's typical for various computers to have specific programs which need this DLL to show the error when you try plus load it up. If you see a certain system show the error, you need to initially uninstall that system, restart a PC plus then resinstall the system again. This must replace the damaged ac1st16.dll file plus remedy the error.<br><br>Maybe you are asking how come these windows XP error messages appear. Well, for we to be capable to know the fix, you must first recognize where those errors come from. There is this software called registry. A registry is software which shops everything on a PC from a normal configuration, setting, info, plus logs of activities from installing to UN-installing, saving to deleting, and a lot more alterations you do in your program pass from it plus gets 'tagged' plus saved because a simple file for healing purposes. Imagine it because a big recorder, a registrar, of all the records in a PC.<br><br>Thus, the number one thing to do when a computer runs slow is to buy an authentic and legal registry repair tool that would enable you eliminate all problems related to registry plus assist you enjoy a smooth running computer. |
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| * polygonal number
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| * a number represented as a discrete r-dimensional regular [[geometry|geometric]] pattern of r-dimensional [[Ball (mathematics)|balls]] such as a [[polygonal number]] (for r = 2) or a '''polyhedral number''' (for r = 3).
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| * a member of the subset of the sets above containing only triangular numbers, pyramidal numbers, and their analogs in other dimensions.<ref>[[Leonard Eugene Dickson|L.E. Dickson]], ''History of the Theory of Numbers''</ref>
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| == Terminology ==
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| Some kinds of figurate number were discussed in the 16th and 17th centuries under the name "figural number".<ref>Simpson, J. A. and Weiner, E. S. C. (Preparers). The Compact Oxford English Dictionary, 2nd ed. Oxford, England: Clarendon Press, 1992, p. 587.</ref>
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| In historical works about [[Greek mathematics]] the preferred term used to be ''figured number''.<ref>A history of Greek Mathematics by [[Thomas Little Heath|T. Heath]]</ref><ref>Greek Mathematical Philosophy by E.A. Maziarz</ref>
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| In a use going back to [[Jakob Bernoulli]]'s [[Ars Conjectandi]],<ref>[[Leonard Eugene Dickson|L.E. Dickson]], ''op. cit.''</ref> the term ''figurate number'' is used for [[triangular]] numbers made up of successive integers, [[tetrahedral number]]s made up of successive triangular numbers, etc. These turn out to be the [[binomial coefficients]]. In this usage the square numbers 4, 9, 16, 25 would not be considered figurate numbers when viewed as arranged in a square.
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| A number of other sources use the term ''figurate number'' as synonymous for the [[polygonal number]]s, either just the usual kind or both those and the [[centered polygonal number]]s.{{Citation needed|date=July 2011}}
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| ==History==
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| The mathematical study of figurate numbers is said to have originated with [[Pythagoras]], possibly based on Babylonian or Egyptian precursors. Generating whichever class of figurate numbers the Pythagoreans studied using [[Gnomon (figure)|gnomons]] is also attributed to Pythagoras. Unfortunately, there is no trustworthy source for these claims, because all surviving writings about the Pythagoreans<ref>The Theoretic Arithmetic of the Pythagoreans, by Thomas Taylor</ref> are from centuries later.<ref>A History of Mathematics, Second Edition, by Carl B. Boyer and Uta C. Merzbach, p. 48</ref> It seems to be certain that the fourth triangular number of ten objects, called [[tetractys]] in Greek, was a central part of the [[Pythagoreanism|Pythagorean religion]], along with several other figures also called tetractys.{{Citation needed|date=July 2011}} Figurate numbers were a concern of Pythagorean geometry.
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| The modern study of figurate numbers goes back to [[Fermat]], specifically the [[Fermat polygonal number theorem]]. Later, it became a significant topic for [[Euler]], who gave an explicit formula for all [[Square triangular number|triangular numbers that are also perfect squares]], among many other discoveries relating to figurate numbers.
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| Figurate numbers have played a significant role in modern recreational mathematics.<ref>Mathematical Recreations, by Maurice Kraitchik</ref> In research mathematics, figurate numbers are studied by way of the [[Ehrhart polynomial]]s, [[polynomial]]s that count the number of integer points in a polygon or polyhedron when it is expanded by a given factor.<ref>{{citation
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| | last1 = Beck | first1 = M.
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| | last2 = De Loera | first2 = J. A.
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| | last3 = Develin | first3 = M.
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| | last4 = Pfeifle | first4 = J.
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| | last5 = Stanley | first5 = R. P. | author5-link = Richard Stanley
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| | contribution = Coefficients and roots of Ehrhart polynomials
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| | location = Providence, RI
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| | mr = 2134759
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| | pages = 15–36
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| | publisher = Amer. Math. Soc.
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| | series = Contemp. Math.
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| | title = Integer points in polyhedra—geometry, number theory, algebra, optimization
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| | volume = 374
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| | year = 2005}}.</ref>
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| ==Triangular numbers==
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| The [[triangular number]]s for n = 1, 2, 3, ... are the result of the juxtaposition of the [[linear number]]s (linear [[gnomon]]s) for n = 1, 2, 3, ...:
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| {| cellpadding="10"
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| |- align="center" valign="bottom"
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| | [[File:GrayDotX.svg|16px|*]]
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| | [[File:GrayDotX.svg|16px|*]] <br>[[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]]
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| | [[File:GrayDotX.svg|16px|*]] <br>[[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]] <br>[[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]]
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| | [[File:GrayDotX.svg|16px|*]] <br>[[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]] <br>[[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]] <br>[[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]]
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| | [[File:GrayDotX.svg|16px|*]] <br>[[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]] <br>[[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]] <br>[[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]] <br>[[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]]
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| | [[File:GrayDotX.svg|16px|*]] <br>[[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]] <br>[[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]] <br>[[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]] <br>[[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]] <br>[[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]][[File:GrayDotX.svg|16px|*]]
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| |}
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| These are the binomial coefficients <math> n+1 \choose 2 </math>. This is the case ''r''=2 of the fact that the ''r''th diagonal of [[Pascal's triangle#Patterns_and_properties|Pascal's triangle]] for <math> r \ge 0 </math> consists of the figurate numbers for the ''r''-dimensional analogs of triangles (''r''-dimensional [[simplex|simplices]]).
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| The simplicial polytopic numbers for r = 1, 2, 3, 4, ... are:
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| *<math>P_1(n) = \frac{n}{1} = {n+0 \choose 1}</math> ([[linear number]]s),
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| *<math>P_2(n) = \frac{n(n+1)}{2} = {n+1 \choose 2}</math> ([[triangular number]]s),
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| *<math>P_3(n) = \frac{n(n+1)(n+2)}{6} = {n+2 \choose 3}</math> ([[tetrahedral number]]s),
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| *<math>P_4(n) = \frac{n(n+1)(n+2)(n+3)}{24} = {n+3 \choose 4}</math> ([[pentachoron number]]s, [[pentatopic number]]s, 4-simplex numbers),
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| <math>...</math>
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| *<math>P_r(n) = \frac{n(n+1)(n+2)...(n+r-1)}{r!} = {n+r-1 \choose r}</math> ([[r-topic number]]s, r-[[simplex]] numbers).
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| The terms ''[[square number]]'' and ''[[cubic number]]'' derive from their geometric representation as a [[Square (geometry)|square]] or [[cube (geometry)|cube]]. The difference of two positive triangular numbers is a [[trapezoidal number]].
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| ==Gnomon==
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| The gnomon is the piece added to a figurate number to transform it to the next larger one.
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| For example, the gnomon of the square number is the [[odd number]], of the general form 2''n'' + 1, ''n'' = 0, 1, 2, 3, ... . The square of size 8 composed of gnomons looks like this:
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| <center>
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| <br>8 8 8 8 8 8 8 8
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| <br>8 7 7 7 7 7 7 7
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| <br>8 7 6 6 6 6 6 6
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| <br>8 7 6 5 5 5 5 5
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| <br>8 7 6 5 4 4 4 4
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| <br>8 7 6 5 4 3 3 3
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| <br>8 7 6 5 4 3 2 2
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| <br>8 7 6 5 4 3 2 1
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| </center>
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| To transform from the ''n-square'' (the square of size ''n'') to the (''n'' + 1)-square, one adjoins 2''n'' + 1 elements: one to the end of each row (''n'' elements), one to the end of each column (''n'' elements), and a single one to the corner. For example, when transforming the 7-square to the 8-square, we add 15 elements; these adjunctions are the 8s in the above figure.
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| This gnomonic technique also provides a [[mathematical proof]] that the sum of the first ''n'' odd numbers is ''n''<sup>2</sup>; the figure illustrates 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 = 64 = 8<sup>2</sup>.
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * {{Citation | last1=Gazalé | first1=Midhat J. | title=Gnomon: From Pharaohs to Fractals | publisher=[[Princeton University Press]] | isbn=978-0-691-00514-0 | year=1999}}
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| * {{Citation | last1=Kraitchik | first1=Maurice | title=Mathematical Recreations: Second Revised Edition | publisher=[[Dover Books]] | isbn=978-0-486-45358-3 | year=2006}}
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| * {{Citation | last1=Deza | first1=Elena | author2=Michel Marie Deza | title=Figurate Numbers, First Edition | publisher=[[World Scientific]] | isbn=978-981-4355-48-3 | year=2012}}
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| * {{Citation | last1=Heath | first1=Thomas Little | title=A history of Greek Mathematics: Volume 1. From Thales to Euclid | publisher=[[Adamant Media Corporation]] | isbn=978-0-543-97448-8 | year=2000}}
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| * {{Citation | last1=Heath | first1=Thomas Little | title=A history of Greek Mathematics: Volume 2. From Aristarchus to Diophantus | publisher=[[Adamant Media Corporation]] | isbn=978-0-543-96877-7 | year=2000}}
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| * {{Citation | last1=Dickson | first1=Leonard Eugene | authorlink = L. E. Dickson | title=[[History of the Theory of Numbers]] (three volume set) | publisher=[[Chelsea Publishing Company, Inc.]] | isbn=ASIN: B000OKO3TK | year=1923}}
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| * {{Citation | last1=Boyer | first1=Carl B. | author2=Uta C. Merzbach | title=A History of Mathematics, Second Edition | publisher= | isbn= | year=}}
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| [[Category:Figurate numbers|*]]
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| [[Category:Integer sequences]]
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"Why does my computer keep freezing up?" I was asked by a lot of individuals the cause of their computer freeze difficulties. And I am fed up with spending much time in answering the query time plus time again. This post is to tell you the real cause of your PC Freezes.
However registry is conveniently corrupted and damaged when you're utilizing your computer. Overtime, without right repair, it usually be loaded with mistakes and incorrect or missing information that usually make a program unable to work properly or apply a certain task. And whenever a system cannot discover the correct information, it usually not learn what to do. Then it freezes up! That is the real cause of your trouble.
If you compare registry cleaners we want a quickly acting registry cleaning. It's no good spending hours plus the PC waiting for the registry cleaning to complete its task. We want your cleaner to complete its task inside minutes.
Always see with it that you have installed antivirus, anti-spyware plus anti-adware programs plus have them updated on a regular basis. This can help stop windows XP running slow.
So to fix this, you really have to be able to make all registry files non-corrupted again. This will dramatically accelerate the loading time of the computer and might allow you to do a large amount of aspects on it again. And fixing these files couldn't be simpler - you only require to utilize a tool called a tuneup utilities.
The initially thing you should do is to reinstall any system which shows the error. It's typical for various computers to have specific programs which need this DLL to show the error when you try plus load it up. If you see a certain system show the error, you need to initially uninstall that system, restart a PC plus then resinstall the system again. This must replace the damaged ac1st16.dll file plus remedy the error.
Maybe you are asking how come these windows XP error messages appear. Well, for we to be capable to know the fix, you must first recognize where those errors come from. There is this software called registry. A registry is software which shops everything on a PC from a normal configuration, setting, info, plus logs of activities from installing to UN-installing, saving to deleting, and a lot more alterations you do in your program pass from it plus gets 'tagged' plus saved because a simple file for healing purposes. Imagine it because a big recorder, a registrar, of all the records in a PC.
Thus, the number one thing to do when a computer runs slow is to buy an authentic and legal registry repair tool that would enable you eliminate all problems related to registry plus assist you enjoy a smooth running computer.