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Square pyramidal number
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==References== {{reflist|refs= <ref name=alsnel>{{citation | last1 = Alsina | first1 = Claudi | last2 = Nelsen | first2 = Roger B. | contribution = Challenge 2.13 | isbn = 978-0-88385-358-0 | location = Washington, DC | mr = 3379535 | pages = 43, 234 | publisher = Mathematical Association of America | series = The Dolciani Mathematical Expositions | title = A Mathematical Space Odyssey: Solid Geometry in the 21st Century | volume = 50 | year = 2015}}</ref> <ref name=anglin>{{citation|last=Anglin|first=W. S. |title=The square pyramid puzzle|jstor=2323911 |journal=[[The American Mathematical Monthly]] |volume=97 |issue=2 |pages=120–124 |year=1990 |doi=10.1080/00029890.1990.11995558}}</ref> <ref name=archimedes>[[Archimedes]], ''[[On Conoids and Spheroids]]'', Lemma to Prop. 2, and ''[[On Spirals]]'', Prop. 10. See {{citation|contribution-url=https://archive.org/details/worksofarchimede00arch/page/106/mode/2up|contribution=Lemma to Proposition 2|title=The Works of Archimedes|year=1897|pages=107–109|url=https://archive.org/details/worksofarchimede00arch|publisher=Cambridge University Press|translator=[[Thomas Heath (classicist)|T. L. Heath]]}}</ref> <ref name=bddps>{{citation | last1 = Beck | first1 = M. | last2 = De Loera | first2 = J. A. | author2-link = Jesús A. De Loera | last3 = Develin | first3 = M. | author3-link = Mike Develin | last4 = Pfeifle | first4 = J. | last5 = Stanley | first5 = R. P. | author5-link = Richard P. Stanley | arxiv = math/0402148 | contribution = Coefficients and roots of Ehrhart polynomials | location = Providence, Rhode Island | mr = 2134759 | pages = 15–36 | series = Contemporary Mathematics | title = Integer Points in Polyhedra—Geometry, Number Theory, Algebra, Optimization | volume = 374 | year = 2005}}</ref> <ref name=beiler>{{citation | author = Beiler, A. H. | title = Recreations in the Theory of Numbers | url = https://archive.org/details/recreationsinthe00beil | url-access = registration | publisher = Dover | year = 1964 | pages = [https://archive.org/details/recreationsinthe00beil/page/194 194–195] | isbn = ((0-486-21096-0))<!-- despite date mismatch, isbn goes to this edition -->}}</ref> <ref name=bvt>{{citation | last1 = Babcock | first1 = Ben | last2 = Van Tuyl | first2 = Adam | arxiv = 1109.5847 | journal = The Australasian Journal of Combinatorics | mr = 3097709 | pages = 77–84 | title = Revisiting the spreading and covering numbers | volume = 56 | year = 2013}}</ref> <ref name=cagbud>{{citation | last1 = Caglayan | first1 = Günhan | last2 = Buddoo | first2 = Horace | date = September 2014 | doi = 10.5951/mathteacher.108.2.0092 | issue = 2 | journal = The Mathematics Teacher | jstor = 10.5951/mathteacher.108.2.0092 | pages = 92–97 | title = Tetrahedral numbers | volume = 108}}</ref> <ref name=conguy>{{citation | last1 = Conway | first1 = John H. | author1-link = John Horton Conway | last2 = Guy | first2 = Richard | author2-link = Richard K. Guy | contribution = Square pyramid numbers | isbn = 978-0-387-97993-9 | pages = 47–49 | publisher = Springer | title = The Book of Numbers | year = 1998}}</ref> <ref name=dpas>{{citation | last1 = Duffin | first1 = Janet | last2 = Patchett | first2 = Mary | last3 = Adamson | first3 = Ann | last4 = Simmons | first4 = Neil | date = November 1984 | issue = 5 | journal = Mathematics in School | jstor = 30216270 | pages = 2–4 | title = Old squares new faces | volume = 13}}</ref> <ref name=fearnehough>{{citation | last = Fearnehough | first = Alan | date = November 2006 | department = Notes | issue = 519 | journal = [[The Mathematical Gazette]] | jstor = 40378200 | pages = 460–461 | title = 90.67 A series for the 'bit' | volume = 90| doi = 10.1017/S0025557200180337 | s2cid = 113711266 | doi-access = free }}</ref> <ref name=federico>{{citation | last = Federico | first = Pasquale Joseph | author-link = Pasquale Joseph Federico | contribution = Pyramidal numbers | doi = 10.1007/978-1-4612-5759-2 | pages = 89–91 | publisher = Springer | series = Sources in the History of Mathematics and Physical Sciences | title = Descartes on Polyhedra: A Study of the "De solidorum elementis" | title-link = Descartes on Polyhedra | volume = 4 | year = 1982| isbn = 978-1-4612-5761-5 }}</ref> <ref name=fibonacci>[[Fibonacci]] (1202), ''[[Liber Abaci]]'', ch. II.12. See {{citation|title=[[Liber Abaci|Fibonacci's Liber Abaci]]|translator=Laurence E. Sigler|publisher=Springer-Verlag|year=2002|isbn=0-387-95419-8|pages=260–261}}</ref> <ref name=grassl>{{citation | last = Grassl | first = Richard | date = July 1995 | doi = 10.2307/3618315 | issue = 485 | journal = [[The Mathematical Gazette]] | jstor = 3618315 | pages = 361–364 | title = 79.33 The squares do fit! | volume = 79| s2cid = 187946568 }}</ref> <ref name=hmu>{{citation |last1=Hopcroft |first1=John E. |author-link1=John Hopcroft |last2=Motwani |first2=Rajeev |author-link2=Rajeev Motwani |last3=Ullman |first3=Jeffrey D. |author-link3=Jeffrey Ullman |title=[[Introduction to Automata Theory, Languages, and Computation]] |edition=3 |year=2007 |publisher=Pearson/Addison Wesley |isbn=9780321455369 |page= [{{Google books|plainurl=y|id=6ytHAQAAIAAJ|page=20}} 20]}}</ref> <ref name=menza>{{citation | last1 = Men'šikov | first1 = G. G. | last2 = Zaezdnyĭ | first2 = A. M. | journal = Žurnal Vyčislitel' noĭ Matematiki i Matematičeskoĭ Fiziki | mr = 196353 | pages = 360–363 | title = Recurrence formulae simplifying the construction of approximating power polynomials | volume = 6 | year = 1966}}; translated into English as {{citation | last1 = Zaezdnyi | first1 = A. M. | last2 = Men'shikov | first2 = G. G. | date = January 1966 | doi = 10.1016/0041-5553(66)90072-3 | issue = 2 | journal = USSR Computational Mathematics and Mathematical Physics | pages = 234–238 | title = Recurrence formulae simplifying the construction of approximating power polynomials | volume = 6}}</ref> <ref name=oeis>{{cite OEIS|A000330|Square pyramidal numbers|mode=cs2}}</ref> <ref name=parker>{{citation | last = Parker | first = Matt | author-link = Matt Parker | contribution = Ship shape | isbn = 978-0-374-53563-6 | mr = 3753642 | pages = 56–59 | publisher = Farrar, Straus and Giroux | location = New York | title = Things to Make and Do in the Fourth Dimension: A Mathematician's Journey Through Narcissistic Numbers, Optimal Dating Algorithms, at Least Two Kinds of Infinity, and More | year = 2015}}</ref> <ref name=robitaille>{{citation | last = Robitaille | first = David F. | date = May 1974 | issue = 5 | journal = The Arithmetic Teacher | jstor = 41190919 | pages = 396–400 | title = Mathematics and chess | volume = 21| doi = 10.5951/AT.21.5.0396 }}</ref> <ref name=stein>{{citation | doi = 10.2307/2688231 | last = Stein | first = Robert G. | title = A combinatorial proof that <math>\textstyle \sum k^3 = (\sum k)^2</math> | journal = [[Mathematics Magazine]] | volume = 44 | issue = 3 | pages = 161–162 | year = 1971 | jstor = 2688231}}</ref> <ref name=yanagihara>{{citation | last = Yanagihara | first = Kitizi | date = November 1918 | issue = 3–4 | journal = [[Tohoku Mathematical Journal]] | pages = 305–324 | title = On the Dajutu or the arithmetic series of higher orders as studied by wasanists | url = https://www.jstage.jst.go.jp/article/tmj1911/14/0/14_0_305/_pdf | volume = 14}}</ref> }}
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