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%I M4619 N1972 #335 Mar 05 2024 08:40:39
%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.
%C 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 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+1,n+1}. - _Eric W. Weisstein_, Jan 02 2018
%C a(n) is the sum of the elements in the multiplication table [0..n] X [0..n]. - _Michel Marcus_, May 06 2021
%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>, Youtube video (2018).
%H Bikash Chakraborty, <a href="https://arxiv.org/abs/2012.11539">Proof Without Words: Sums of Powers of Natural numbers</a>, arXiv:2012.11539 [math.HO], 2020.
%H Robert Dawson, <a href="https://www.emis.de/journals/JIS/VOL21/Dawson/dawson6.html">On Some Sequences Related to Sums of Powers</a>, J. Int. Seq., Vol. 21 (2018), Article 18.7.6.
%H Shel Kaphan, <a href="/A000537/a000537_4.txt">How to see that the difference between two successive squared triangular numbers is a cube</a>
%H Sameen Ahmed Khan, <a href="https://doi.org/10.12732/ijam.v33i2.6">Sums of the powers of reciprocals of polygonal numbers</a>, Int'l J. of Appl. Math. (2020) Vol. 33, No. 2, 265-282.
%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&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="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 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'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, row sums of A127777. - _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 Sum_{k > 0} 1/a(k) = (4/3)*(Pi^2-9). - _Jaume Oliver Lafont_, Sep 20 2009
%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) = 6*A000332(n+2) + A000217(n), (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
%F Another identity: ..., a(3) = (1/2)*(1*(2+4+6)+3*(4+6)+5*6) = 36, a(4) = (1/2)*(1*(2+4+6+8)+3*(4+6+8)+5*(6+8)+7*(8)) = 100, a(5) = (1/2)*(1*(2+4+6+8+10)+3*(4+6+8+10)+5*(6+8+10)+7*(8+10)+9*(10)) = 225, ... - _J. M. Bergot_, Aug 27 2022
%F Comment from _Michael Somos_, Aug 28 2022: (Start)
%F The previous comment expresses a(n) as the sum of all of the n X n multiplication table array entries.
%F For example, for n = 4:
%F 1 2 3 4
%F 2 4 6 8
%F 3 6 9 12
%F 4 8 12 16
%F This array sum can be split up as follows:
%F +---+---------------+
%F | 0 | 1 2 3 4 | (0+1)*(1+2+3+4)
%F | +---+-----------+
%F | 0 | 2 | 4 6 8 | (1+2)*(2+3+4)
%F | | +---+-------+
%F | 0 | 3 | 6 | 9 12 | (2+3)*(3+4)
%F | | | +---+---+
%F | 0 | 4 | 8 |12 |16 | (3+4)*(4)
%F +---+---+---+---+---+
%F This kind of row+column sums was used by Ramanujan and others for summing Lambert series. (End)
%e G.f. = x + 9*x^2 + 36*x^3 + 100*x^4 + 225*x^5 + 441*x^6 + ... - _Michael Somos_, Aug 29 2022
%p a:= n-> (n*(n+1)/2)^2:
%p seq(a(n), n=0..40);
%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
%o (GAP) List([0..40],n->(n*(n+1)/2)^2); # _Muniru A Asiru_, Dec 05 2018
%o (Python)
%o def A000537(n): return (n*(n+1)>>1)**2 # _Chai Wah Wu_, Oct 20 2023
%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.
%Y Cf. A000332, A000566, A035287, A039623, A053382, A053383, A059376, A059827, A059860, A085582, A127777, A176271.
%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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