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A000537
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Sum of first n cubes; or n-th triangular number squared.
(Formerly M4619 N1972)
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104
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0, 1, 9, 36, 100, 225, 441, 784, 1296, 2025, 3025, 4356, 6084, 8281, 11025, 14400, 18496, 23409, 29241, 36100, 44100, 53361, 64009, 76176, 90000, 105625, 123201, 142884, 164836, 189225, 216225, 246016, 278784, 314721, 354025, 396900, 443556, 494209, 549081
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OFFSET
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0,3
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COMMENTS
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Number of parallelograms in an n X n rhombus - Matti De Craene (Matti.DeCraene(AT)rug.ac.be), May 14 2000.
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.
Also number of 2-dimensional cage assemblies (cf. A059827, A059860).
The n-th triangular number T(n)=sum_r(1, n)=n(n+1)/2 satisfies the relations: (i) T(n) + T(n-1)=n^2 and (ii) T(n) - T(n-1)=n from definition, so that n^2*n=n^3={T(n)}^2 - {T(n-1)}^2 and thus summing telescopingly over 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
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.
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.
1^3 + 2^3 + 3^3 +...+ n^3=(1+2+3+...+n)^2
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
Also number of hexagons with vertices in an hexagonal grid with n points in each side. - Ignacio Larrosa Canestro (ilarrosa(AT)mundo-r.com), Oct 15 2006
Number of permutations of n distinct letters (ABCD...) each of which appears twice with 4 and n-4 fixed points. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 09 2006
With offset 1 = binomial transform of [1, 8, 19, 18, 6,...]. [Gary W. Adamson, Dec 03 2008]
Sum(k>0,1/a(k))=(4/3)*(Pi^2-9). [Jaume Oliver Lafont, Sep 20 2009]
a(n) = SUM(A176271(m,k)): 1<=k<=m<=n). [Reinhard Zumkeller, Apr 13 2010]
The sequence is related to A000330 by a(n) = n*A000330(n)-sum(A000330(i), i=0..n-1): this is the case d=1 in the identity n*(n*(d*n-d+2)/2)-sum(i*(d*i-d+2)/2, i=0..n-1) = n*(n+1)*(2*d*n-2*d+3)/6. - Bruno Berselli, Apr 26 2010, Mar 01 2012
From Wolfdieter Lang, Jan 11 2013: (Start)
For sums of powers of positive integers S(k,n):=sum(j^k,j=1..n) one has the recurrence S(k,n) = (n+1)*S(k-1,n) - sum(S(k-1,l),l=1..n), n >= 1, k >= 1.
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.
This trick generalizes immediately to arbitrary powers k. For k=3: a(n) = (n+1)*A000330(n) - sum(A000330(l),l=1..n), which coincides with the formula given in the preceding Berselli comment. (End)
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REFERENCES
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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.
A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 110ff.
Marcel Berger, Encounter with a Geometer, Part II, Notices of the American Mathematical Society, Vol. 47, No. 3, (March 2000), pp. 326-340. [About the work of Mikhael Gromov].
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 155.
John H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, pp. 36, 58.
Clifford Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Oxford University Press, 2001, p. 325.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
H. K. Strick, Geschichten aus der Mathematik II, Spektrum Spezial 3/11, p. 13.
D. Wells, You Are A Mathematician, "Counting rectangles in a rectangle", Problem 8H, pp. 240; 254, Penguin Books 1995.
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LINKS
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T. D. Noe, Table of n, a(n) for n = 0..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
M. Azaola and F. Santos, The number of triangulations of the cyclic polytope C(n,n-4), Discrete Comput. Geom., 27 (2002), 29-48 (see Prop. 4.2(b)).
B. Berselli, A description of the recursive method in Comments lines: website Matem@ticamente (in Italian).
Wolfdieter Lang, Ibn al-Haytham's trick.
C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review
_Simon Plouffe_, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
_Simon Plouffe_, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Eric Weisstein's World of Mathematics, Faulhaber's Formula
Wikipedia, Faulhaber's formula
G. Xiao, Sigma Server, Operate on "n^3"
Index entries for sequences related to linear recurrences with constant coefficients
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FORMULA
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a(n) = (n*(n+1)/2)^2 = A000217(n)^2, that is, 1^3 + 2^3 + 3^3 +...+ n^3 = (1+2+3+...+n)^2.
G.f.: (x+4*x^2+x^3)/(1-x)^5.
a(n) = Sum [ Sum ( 1 + Sum (6*n) ) ]. - Xavier Acloque, Jan 21 2003
triangle(n)*Sum(j=1, n, j) - Jon Perry, Jul 28 2003
a(n) = Sum[Sum[(i*j), {i, 1, n}], {j, 1, n}] - Alexander Adamchuk, Oct 24 2004
a(n)=A035287(n)/4 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 09 2007
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
G.f.: x*F(3,3;1;x); [From Paul Barry, Sep 18 2008]
a(n) = sum(i = 1..n, J_3(i)*floor(n/i)), where J_ 3 is A059376. - Enrique Pérez Herrero, Feb 26 2012
a(n) = sum_{i=1..n} sum_{j=1..n} sum_{k=1..n} max(i,j,k). - Enrique Pérez Herrero, Feb 26 2013
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MAPLE
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(n*(n+1)/2)^2;
A000537:=-(1+4*z+z**2)/(z-1)**5; [Simon Plouffe in his 1992 dissertation for sequence without initial zero.]
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MATHEMATICA
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Table[ Sum[(i*j), {i, n}, {j, n}], {n, 0, 38}] (* modified by Robert G. Wilson v, Nov 16 2012 *)
Accumulate[Range[0, 50]^3] (* Harvey P. Dale, Mar 1 2011 *)
f[n_] := n^2 (n + 1)^2/4; Array[f, 39, 0] (* Robert G. Wilson v, Nov 16 2012 *)
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PROG
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(PARI) a(n)=(n*(n+1)/2)^2
(PARI) t(n)=n*(n+1)/2 for(i=1, 30, print1(", "sum(j=1, i, j*t(i))))
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CROSSREFS
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Convolution of A000217 and A008458. Cf. A000330, A006003, A000538.
Row sums of triangles A094414 and A094415.
Second column of triangle A008459.
Row 3 of array A103438.
Cf. A002415.
Cf. A101102, A101097, A101094, A024166, A000578.
Sequence in context: A187607 A085037 A169835 * A114286 A098928 A139469
Adjacent sequences: A000534 A000535 A000536 * A000538 A000539 A000540
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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