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A000537 Sum of first n cubes; or n-th triangular number squared.
(Formerly M4619 N1972)
153

%I M4619 N1972

%S 0,1,9,36,100,225,441,784,1296,2025,3025,4356,6084,8281,11025,14400,

%T 18496,23409,29241,36100,44100,53361,64009,76176,90000,105625,123201,

%U 142884,164836,189225,216225,246016,278784,314721,354025,396900,443556,494209,549081

%N Sum of first n cubes; or n-th triangular number squared.

%C Number of parallelograms in an n X n rhombus. - Matti De Craene (Matti.DeCraene(AT)rug.ac.be), May 14 2000

%C Or, number of orthogonal rectangles in an n X n checkerboard, or rectangles in an n X n array of squares. - _Jud McCranie_, Feb 28 2003. Compare A085582.

%C Also number of 2-dimensional cage assemblies (cf. A059827, A059860).

%C The n-th triangular number T(n) = Sum_{r=1..n} r = n(n+1)/2 satisfies the relations: (i) T(n) + T(n-1) = n^2 and (ii) T(n) - T(n-1) = n by definition, so that n^2*n = n^3 = {T(n)}^2 - {T(n-1)}^2 and by summing on n we have Sum_{ r = 1..n } r^3 = {T(n)}^2 = (1+2+3+...+n)^2 = (n*(n+1)/2)^2. - _Lekraj Beedassy_, May 14 2004

%C Number of 4-tuples of integers from {0,1,...,n}, without repetition, whose last component is strictly bigger than the others. Number of 4-tuples of integers from {1,...,n}, with repetition, whose last component is greater than or equal to the others.

%C Number of ordered pairs of two element subsets of {0,1,...,n} without repetition. Number of ordered pairs of 2-element multisubsets of {1,...,n} with repetition.

%C 1^3 + 2^3 + 3^3 + ... + n^3 = (1 + 2 + 3 + ... + n)^2.

%C a(n) is the number of parameters needed in general to know the Riemannian metric g of an n-dimensional Riemannian manifold (M,g), by knowing all its second derivatives; even though to know the curvature tensor R requires (due to symmetries) (n^2)*(n^2-1)/12 parameters, a smaller number (and a 4-dimensional pyramidal number). - _Jonathan Vos Post_, May 05 2006

%C Also number of hexagons with vertices in an hexagonal grid with n points in each side. - _Ignacio Larrosa Cañestro_, Oct 15 2006

%C Number of permutations of n distinct letters (ABCD...) each of which appears twice with 4 and n-4 fixed points. - _Zerinvary Lajos_, Nov 09 2006

%C With offset 1 = binomial transform of [1, 8, 19, 18, 6, ...]. - _Gary W. Adamson_, Dec 03 2008

%C Sum_{k > 0} 1/a(k) = (4/3)*(Pi^2-9). - _Jaume Oliver Lafont_, Sep 20 2009

%C a(n) = Sum_{1 <= k <= m <= n} A176271(m,k). - _Reinhard Zumkeller_, Apr 13 2010

%C The sequence is related to A000330 by a(n) = n*A000330(n) - Sum_{i=0..n-1} A000330(i): this is the case d=1 in the identity n*(n*(d*n-d+2)/2) - Sum_{i=0..n-1} i*(d*i-d+2)/2 = n*(n+1)*(2*d*n-2*d+3)/6. - _Bruno Berselli_, Apr 26 2010, Mar 01 2012

%C From _Wolfdieter Lang_, Jan 11 2013: (Start)

%C For sums of powers of positive integers S(k,n) := Sum_{j=1..n}j^k one has the recurrence S(k,n) = (n+1)*S(k-1,n) - Sum_{l=1..n} S(k-1,l), n >= 1, k >= 1.

%C This was used for k=4 by Ibn al-Haytham in an attempt to compute the volume of the interior of a paraboloid. See the Strick reference where the trick he used is shown, and the W. Lang link.

%C This trick generalizes immediately to arbitrary powers k. For k=3: a(n) = (n+1)*A000330(n) - Sum_{l=1..n} A000330(l), which coincides with the formula given in the previous comment by Berselli. (End)

%C Regarding to the previous contribution, see also Matem@ticamente in Links field and comments on this recurrences in similar sequences (partial sums of n-th powers). - _Bruno Berselli_, Jun 24 2013

%C A rectangular prism with sides A000217(n), A000217(n+1), and A000217(n+2) has surface area 6*a(n+1). - _J. M. Bergot_, Aug 07 2013, edited with corrected indices by _Antti Karttunen_, Aug 09 2013

%C A formula for the r-th successive summation of k^3, for k = 1 to n, is (6*n^2+r*(6*n+r-1)*(n+r)!)/((r+3)!*(n-1)!),(H. W. Gould). - _Gary Detlefs_, Jan 02 2014

%C Note that this sequence and its formula were known to (and possibly discovered by) Nicomachus, predating Ibn al-Haytham by 800 years. - _Charles R Greathouse IV_, Apr 23 2014

%C a(n) is the number of ways to paint the sides of a nonsquare rectangle using at most n colors. Cf. A039623. - _Geoffrey Critzer_, Jun 18 2014

%C For n > 0: A256188(a(n)) = A000217(n) and A256188(m) != A000217(n) for m < a(n), i.e., positions of first occurrences of triangular numbers in A256188. - _Reinhard Zumkeller_, Mar 26 2015

%C There is no cube in this sequence except 0 and 1. - _Altug Alkan_, Jul 02 2016

%C Also the number of chordless cycles in the complete bipartite graph K_n,n. - _Eric W. Weisstein_, Jan 02 2018

%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 813.

%D Avner Ash and Robert Gross, Summing it up, Princeton University Press, 2016, p. 62, eq. (6.3) for k=3.

%D A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 110ff.

%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 155.

%D John H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, pp. 36, 58.

%D Clifford Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Oxford University Press, 2001, p. 325.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D H. K. Strick, Geschichten aus der Mathematik II, Spektrum Spezial 3/11, p. 13.

%D D. Wells, You Are A Mathematician, "Counting rectangles in a rectangle", Problem 8H, pp. 240; 254, Penguin Books 1995.

%H T. D. Noe, <a href="/A000537/b000537.txt">Table of n, a(n) for n = 0..1000</a>

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H Luciano Ancora, <a href="/A000578/a000578.pdf">Sum of cubes of the first "n" natural numbers</a>

%H Luciano Ancora, <a href="https://upload.wikimedia.org/wikipedia/commons/9/9c/FigurateN.pdf">The Square Pyramidal Number and other figurate numbers</a>

%H M. Azaola and F. Santos, <a href="http://personales.unican.es/santosf/Articulos/">The number of triangulations of the cyclic polytope C(n,n-4)</a>, Discrete Comput. Geom., 27 (2002), 29-48 (see Prop. 4.2(b)).

%H Marcel Berger, <a href="http://www.ams.org/notices/200003/fea-berger.pdf">Encounter with a Geometer, Part II</a>, Notices of the American Mathematical Society, Vol. 47, No. 3, (March 2000), pp. 326-340. [About the work of Mikhael Gromov.]

%H B. Berselli, A description of the recursive method in Comments lines: website <a href="http://www.lanostra-matematica.org/2008/12/sequenze-numeriche-e-procedimenti.html">Matem@ticamente</a> (in Italian).

%H blackpenredpen, <a href="https://www.youtube.com/watch?v=Uq9OXC0Gzgw">Math for fun, how many rectangles?</a> (2018)

%H Wolfdieter Lang, <a href="/A000537/a000537.pdf">Ibn al-Haytham's trick.</a>

%H S. Legendre, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Legendre/legendre2.html">The Number of Crossings in a Regular Drawing of the Complete Bipartite Graph </a>, JIS 12 (2009) 09.5.5

%H Henri Picciotto, <a href="http://www.mathedpage.org/infinity/cubes.html">Sum of Cubes</a>, Proof without words.

%H C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," <a href="http://www.zentralblatt-math.org/zmath/en/search/?q=an:0983.00008&amp;format=complete">Zentralblatt review</a>

%H C. J. Pita Ruiz V., <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Pita/pita19.html">Some Number Arrays Related to Pascal and Lucas Triangles</a>, J. Int. Seq. 16 (2013) #13.5.7

%H Simon Plouffe_, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992.

%H Simon Plouffe_, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">1031 Generating Functions and Conjectures</a>, Université du Québec à Montréal, 1992.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ChordlessCycle.html">Chordless Cycle</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CompleteBipartiteGraph.html">Complete Bipartite Graph</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FaulhabersFormula.html">Faulhaber's Formula</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Faulhaber&#39;s_formula">Faulhaber's formula</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Squared_triangular_number">Squared triangular number</a>

%H G. Xiao, Sigma Server, <a href="http://wims.unice.fr/~wims/en_tool~analysis~sigma.en.html">Operate on "n^3"</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

%F a(n) = (n*(n+1)/2)^2 = A000217(n)^2 = Sum_{k=1..n} A000578(k), that is, 1^3 + 2^3 + 3^3 + ... + n^3 = (1 + 2 + 3 + ... + n)^2.

%F G.f.: (x+4*x^2+x^3)/(1-x)^5. - _Simon Plouffe_ in his 1992 dissertation

%F a(n) = Sum ( Sum ( 1 + Sum (6*n) ) ), rephrasing the formula in A000578. - Xavier Acloque, Jan 21 2003

%F a(n) = Sum_{i=1..n} Sum_{j=1..n} i*j. - _Alexander Adamchuk_, Oct 24 2004

%F a(n) = A035287(n)/4. - _Zerinvary Lajos_, May 09 2007

%F This sequence could be obtained from the general formula n*(n+1)*(n+2)*(n+3)*...*(n+k)*(n*(n+k) + (k-1)*k/6)/((k+3)!/6) at k=1. - _Alexander R. Povolotsky_, May 17 2008

%F G.f.: x*F(3,3;1;x). - _Paul Barry_, Sep 18 2008

%F a(n) = Sum_{i=1..n} J_3(i)*floor(n/i), where J_ 3 is A059376. - _Enrique Pérez Herrero_, Feb 26 2012

%F a(n) = Sum_{i=1..n} Sum_{j=1..n} Sum_{k=1..n} max(i,j,k). - _Enrique Pérez Herrero_, Feb 26 2013

%F a(n) = 6*C(n+2,4) + C(n+1,2), (Knuth). - _Gary Detlefs_, Jan 02 2014

%F a(n) = -Sum_{j=1..3} j*s(n+1,n+1-j)*S(n+3-j,n), where s(n,k) and S(n,k) are the Stirling numbers of the first kind and the second kind, respectively. - _Mircea Merca_, Jan 25 2014

%F Sum_{n>=1} (-1)^(n+1)/a(n) = 4*(3-4*log(2)). - _Vaclav Kotesovec_, Feb 13 2015

%F a(n)*((s-2)*(s-3)/2) = P(3, P(s, n+1)) - P(s, P(3, n+1)), where P(s, m) = ((s-2)*m^2-(s-4)*m)/2 is the m-th s-gonal number. For s=7, 10*a(n) = A000217(A000566(n+1)) - A000566(A000217(n+1)). - _Bruno Berselli_, Aug 04 2015

%F From _Ilya Gutkovskiy_, Jul 03 2016: (Start)

%F E.g.f.: x*(4 + 14*x + 8*x^2 + x^3)*exp(x)/4.

%F Dirichlet g.f.: (zeta(s-4) + 2*zeta(s-3) + zeta(s-2))/4. (End)

%F a(n) = (Bernoulli(4, n+1) - Bernoulli(4, 1))/4, n >= 0, with the Bernoulli polynomial B(4, x) from row n=4 of A053382/A053383. See, e.g., the Ash-Gross reference, p. 62, eq. (6.3) for k=3. - _Wolfdieter Lang_, Mar 12 2017

%F a(n) = A000217((n+1)^2) - A000217(n+1)^2. - _Bruno Berselli_, Aug 31 2017

%F a(n) = n*binomial(n+2, 3) + binomial(n+2, 4) + binomial(n+1, 4). - _Tony Foster III_, Nov 14 2017

%p (n*(n+1)/2)^2;

%t Accumulate[Range[0, 50]^3] (* _Harvey P. Dale_, Mar 01 2011 *)

%t f[n_] := n^2 (n + 1)^2/4; Array[f, 39, 0] (* _Robert G. Wilson v_, Nov 16 2012 *)

%t Table[CycleIndex[{{1, 2, 3, 4}, {3, 2, 1, 4}, {1, 4, 3, 2}, {3, 4, 1, 2}}, s] /. Table[s[i] -> n, {i, 1, 2}], {n, 0, 30}] (* _Geoffrey Critzer_, Jun 18 2014 *)

%t Accumulate @ Range[0, 50]^2 (* _Waldemar Puszkarz_, Jan 24 2015 *)

%t Binomial[Range[20], 2]^2 (* _Eric W. Weisstein_, Jan 02 2018 *)

%t LinearRecurrence[{5, -10, 10, -5, 1}, {0, 1, 9, 36, 100}, 20] (* _Eric W. Weisstein_, Jan 02 2018 *)

%t CoefficientList[Series[-((x (1 + 4 x + x^2))/(-1 + x)^5), {x, 0, 20}], x] (* _Eric W. Weisstein_, Jan 02 2018 *)

%o (PARI) a(n)=(n*(n+1)/2)^2

%o (MAGMA) [(n*(n+1)/2)^2: n in [0..50]]; // _Wesley Ivan Hurt_, Jun 06 2014

%o (Haskell) a000537 = a000290 . a000217 -- _Reinhard Zumkeller_, Mar 26 2015

%Y Convolution of A000217 and A008458.

%Y Cf. A000330, A000538, A006003.

%Y Row sums of triangles A094414 and A094415.

%Y Second column of triangle A008459.

%Y Row 3 of array A103438.

%Y Cf. A000578, A002415, A024166, A101102, A101094, A101097.

%Y Cf. A236770 (see crossrefs).

%Y Cf. A000290, A253169, A256188.

%K nonn,easy,nice

%O 0,3

%A _N. J. A. Sloane_

%E Edited by _M. F. Hasler_, May 02 2015

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Last modified June 24 23:40 EDT 2018. Contains 311877 sequences. (Running on oeis4.)