en>Spacerock1: /* Calculation from state vectors */ Edited vector format

2013-12-31T16:04:02Z

Calculation from state vectors: Edited vector format

← Older revision		Revision as of 18:04, 31 December 2013
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	~~Nice~~ to ~~meet you~~, ~~I am Marvella Shryock~~. ~~My day occupation~~ is ~~a meter reader~~. ~~Years ago we moved to North Dakota~~. ~~Doing ceramics~~ is ~~what love performing~~.<br><br>~~Also visit my page: [~~http://www.~~Tobeycmossgallery~~.~~com~~/~~dietmealdelivery74619 http:~~//~~www~~.~~Tobeycmossgallery~~.~~com~~]		A '''[[pentatope]] number''' is a number in the fifth cell of any row of [[Pascal's triangle]] starting with the 5-term row 1 4 6 4 1 either from left to right or from right to left.

			The first few numbers of this kind are :

			: [[1 (number)\|1]], [[5 (number)\|5]], [[15 (number)\|15]], [[35 (number)\|35]], [[70 (number)\|70]], [[126 (number)\|126]], 210, 330, [[495 (number)\|495]], 715, [[1001 (number)\|1001]], 1365 {{OEIS\|id=A000332}}

			[[Image:Pentatope of 70 spheres animation.gif\|frame\|right\|A [[pentatope]] with side length 5 contains 70 spheres. Each layer represents one of the first five [[tetrahedral number]]s. For example the bottom (green) layer has 35 spheres in total.]]
			Pentatope numbers belong in the class of [[figurate number]]s, which can be represented as regular, discrete geometric patterns.<ref>{{citation\|title=Figurate Numbers\|first1=Elena\|last1=Deza\|first2=M.\|last2=Deza\|author2-link=Michel Deza\|publisher=World Scientific\|year=2012\|isbn=9789814355483\|page=162\|contribution=3.1 Pentatope numbers and their multidimensional analogues}}</ref> The formula for the ''n''th pentatopic number is:

			: <math>{n + 3 \choose 4} = \frac{n(n+1)(n+2)(n+3)}{24} = {n^{\overline 4} \over 4!}.</math>

			Two of every three pentatope numbers are also [[pentagonal number]]s. To be precise, the (3''k'' − 2)th pentatope number is always the ((3''k''<sup>2</sup> − ''k'')/2)th pentagonal number and the (3''k'' − 1)th pentatope number is always the ((3''k''<sup>2</sup> + ''k'')/2)th pentagonal number. The 3''k''th pentatope number is the [[pentagonal number\|generalized pentagonal number]] obtained by taking the negative index −(3''k''<sup>2</sup> + ''k'')/2 in the formula for pentagonal numbers. (These expressions always give integers).<ref name="oeis">{{SloanesRef\|A000332}}</ref>

			The infinite sum of the reciprocals of all pentatopal numbers is <math>4 \over 3</math>.<ref>{{citation\|title=Sums of the inverses of binomial coefficients\|journal=Fibonacci Quarterly\|year=1981\|url=http://www.fq.math.ca/Scanned/19-5/rockett.pdf\|first=Andrew M.\|last=Rockett\|volume=19\|issue=5\|pages=433–437}}. Theorem 2, p. 435.</ref> This can be derived using [[telescoping series]].

			: <math> \sum_{n=1}^\infty {4! \over {n(n+1)(n+2)(n+3)}} = {4 \over 3} </math>

			Pentatopal numbers can also be represented as the sum of the first n [[tetrahedral number]]s.<ref name="oeis"/>

			==References==
			{{reflist}}

			{{Classes of natural numbers}}
			{{Num-stub}}

			[[Category:Figurate numbers]]

en>EmausBot: r2.7.2+) (Robot: Modifying ar:زاوية العقدة المدارية

2012-08-13T01:03:59Z

r2.7.2+) (Robot: Modifying ar:زاوية العقدة المدارية

New page

Nice to meet you, I am Marvella Shryock. My day occupation is a meter reader. Years ago we moved to North Dakota. Doing ceramics is what love performing.<br><br>Also visit my page: [http://www.Tobeycmossgallery.com/dietmealdelivery74619 http://www.Tobeycmossgallery.com]

Longitude of the ascending node - Revision history

en>Spacerock1: /* Calculation from state vectors */ Edited vector format

en>EmausBot: r2.7.2+) (Robot: Modifying ar:زاوية العقدة المدارية