0,3

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 (blekraj(AT)yahoo.com), 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 (jvospost3(AT)gmail.com), 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,...]. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 03 2008]

Sum(k>0,1/a(k))=(4/3)*(Pi^2-9) [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Sep 20 2009]

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

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

S. 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.

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.

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

T. D. Noe, Table of n, a(n) for n=0..1000

Index entries for sequences related to linear recurrences with constant coefficients

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].

S. 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.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

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)).

C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review

G. Xiao, Sigma Server, Operate on "n^3"

a(n) = (n*(n+1)/2)^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

Sum(j=1, n, j*triangle(n)) - Jon Perry (perry(AT)globalnet.co.uk), Jul 28 2003

a(n) = Sum[Sum[(i*j), {i, 1, n}], {j, 1, n}] - Alexander Adamchuk (alex(AT)kolmogorov.com), Oct 24 2004

Equals A000217(n)^2. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 26 2007

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 (pevnev(AT)juno.com), May 17 2008

G.f.: x*F(3,3;1;x); [From Paul Barry (pbarry(AT)wit.ie), Sep 18 2008]

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

[seq((binomial(n, 2))^2, n=1..35)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 24 2006

[seq (stirling2(n+1, n)*binomial(n+1, 2), n=0..34)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 06 2006

a:=array(0...34): a[0]:=0: a[1]:=1:print(0, a[0]); print(1, a[1]); for i from 2 to 34 do a[i]:= a[i-1]+(i^3):print(i, a[i]); od: - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 26 2007

a:=n->sum(sum(n^2/4, j=0..n), k=0..n): seq(a(n), n=0..34); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 09 2007

seq(sum(sum(lcm(k, j), j=1..n), k=0..n), n=0..34); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 01 2007

A000537:=-(1+4*z+z**2)/(z-1)**5; [S. Plouffe in his 1992 dissertation for sequence without initial zero.]

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=a[n-1]+n^3 od: seq(a[n], n=0..34); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 22 2008

a:=n->sum(k*sum(k, k=0..n), k=0..n):seq(a(n), n=0...34); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Aug 01 2008

a:=n->sum(k*sum(k, k=0..n), k=0..n):seq(a(n), n=0...34); [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Aug 09 2008]

Table[Sum[Sum[(i*j), {i, 1, n}], {j, 1, n}], {n, 0, 10}]

Table[StirlingS2[i+1, i]*(-StirlingS1[i+1, i]), {i, 0, 34}] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 24 2007

s = 0; lst = {s}; Do[s += n^3; AppendTo[lst, s], {n, 1, 42, 1}]; lst [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 12 2009]

(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))))

(PARI) a(n)=sum(m=1, n, sum(i=m*(m+1)/2-m+1, m*(m+1)/2, (2*i-1))) - Alexander R. Povolotsky (pevnev(AT)juno.com), Nov 05 2007

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. A000217, A002415.

Cf. A101102, A101097, A101094, A024166, A000578.

Sequence in context: A134537 A066647 A085037 this_sequence A114286 A098928 A139469

Adjacent sequences: A000534 A000535 A000536 this_sequence A000538 A000539 A000540

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).

Removed attribute "conjectured" from Plouffe g.f R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 11 2009