0,2

(1) a(n) = sum of second string of n triangular numbers - sum of first n triangular numbers, or the 2n-th partial sum of triangular numbers (A000217 ) - the n-th partial sum of triangular numbers(A000217 ). The same for natural numbers gives squares. (2) a(n) = (n-th triangular number)*(the n-th even number) = n(n+1)/2 * (2n) - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Nov 05 2002

Let M(n) be the n X n matrix m(i,j)=1/(i+j+x), let P(n,x)=prod(i=0,n-1,i!^2)/det(M(n)). Then P(n,x) is a polynomial with integer coefficients of degree n^2 and a(n) is the coefficient of x^(n^2-1). - Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 15 2003

The sequence allows us to find Y values of the equation: (X-Y)^3-XY=0. Sequence gives Y values. To find X values: a(n)=n*(n+1)^2. (see A045991) - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), May 09 2006

a(2d-1) is the number of self-avoiding walk of length 3 in the d-dimensional hypercubic lattice. - Michael Somos Sep 06 2006

1/2 + 1/12 + 1/36...=((Pi)^2 - 6)/6. [Jolley] - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 20 2006

1/2 + 1/12 + 1/36 +...= (Pi^2 - 6)/6 [Jolley] - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 22 2006

Number of units of a(n) belongs to a periodic sequence: 0, 2, 2, 6, 0.We conclude that a(n) and a(n+5) have the same number of units. [From Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Sep 05 2009]

L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, pp. 50, 64.

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

a(n) = Sum[Sum[(i+j), {i, 1, n}], {j, 1, n}] - Alexander Adamchuk (alex(AT)kolmogorov.com), Oct 12 2004

a(n)sum(sum(n, j=0..n),k=1..n), n>=0 . - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 11 2007

a(3)=3^2+3^3=36.

a:=n->sum(n*numbperm (n, 1), j=0..n): seq(a(n), n=0..39); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 25 2007

a:=n->sum(sum(n, j=0..n), k=1..n): seq(a(n), n=0..39); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 11 2007

a:=n->add(add(add(1, j=1..n), j=2..n), j=2..n):seq(a(n), n=1..21); [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Aug 27 2008]

a:=n->sum ((j*(n+1))+(j*(n-1)), j=0..n): seq(a(n), n=0..39); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 18 2009]

a[n_]:=n^2; b[n_]:=n^3; lst={}; Do[AppendTo[lst, b[n]+a[n]], {n, 0, 6!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Jan 03 2009]

Cf. A022549. Twice A002411.

Cf. A045991.

Sequence in context: A055707 A055699 A062094 this_sequence A073404 A141208 A035597

Adjacent sequences: A011376 A011377 A011378 this_sequence A011380 A011381 A011382

nonn,easy

Glen Burch; Felice Russo (felice.russo(AT)katamail.com)