(1) a(n) = sum of second string of n triangular numbers  sum of first n triangular numbers, or the 2nth partial sum of triangular numbers (A000217 )  the nth partial sum of triangular numbers(A000217 ). The same for natural numbers gives squares. (2) a(n) = (nth triangular number)*(the nth even number) = n(n+1)/2 * (2n)  Amarnath Murthy, 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,n1,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^21).  Benoit Cloitre, Jan 15 2003
Y values of solutions of the equation: (XY)^3X*Y=0. X values are a(n)=n*(n+1)^2 (see A045991)  Mohamed Bouhamida (bhmd95(AT)yahoo.fr), May 09 2006
sum_{n>0} 1/a(n) = (Pi^2  6)/6 = 0.6449340.. [Jolley eq 272]  Gary W. Adamson, Dec 22 2006
a(2d1) is the number of selfavoiding walk of length 3 in the ddimensional hypercubic lattice.  Michael Somos, Sep 06 2006
a(n) mod 10 is periodic 5: repeat [0, 2, 2, 6, 0]. [Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Sep 05 2009]
This sequence is related to A005449 by a(n) = n*A005449(n)sum(A005449(i), i=0..n1), and this is the case d=3 in the identity n^2*(d*n+d2)/2sum(k*(d*k+d2)/2, k=0..n1) = n*(n+d)*(2*d*n+d3)/6.  Bruno Berselli, Nov 18 2010
Using (n, n+1) to generate a primitive Pythagorean triangle, the sides will be 2*n+1, 2*(n^2+n), and 2*n^2+2*n+1. Inscribing the largest rectangle with integral sides will have sides of length n and n^2+n. Side n is collinear to side 2*n+1 of the triangle and side n^2+n is collinear to side 2*(n^2+n) of the triangle. The areas of theses rectangles are a(n). [J. M. Bergot, Sep 22 2011]
a(n+1) = sum of nth row of the triangle in A195437.  Reinhard Zumkeller, Nov 23 2011
Partial sums of A049450.  Omar E. Pol, Jan 12 2013
From Jon Perry, May 11 2013: (Start)
Define a 'stable brick triangle' as:

 c 

 a   b 

with a, b, c > 0 and c <= a + b. This can be visualized as two bricks with a third brick on top. The third brick can only be as strong as a+b, otherwise the wall collapses  for example, (1,2,4) is unstable.
a(n) gives the number of stable brick triangles that can be formed if the two supporting bricks are 1<=a<=n and 1<=b<=n: a(n) = sum_{a=1..n} sum_{b=1..n} sum_c 1 = n^3+n^2 as given in the Adamchuk formula.
So for i=j=n=2 we have 4:
1 2 3 4
2 2 2 2 2 2 2 2
For example, n=2 gives 2 from [a=1,b=1], 3 from both [a=1,b=2] and [a=2,b=1] and 4 from [a=2,b=2] so a(2) = 2 + 3 + 3 + 4 = 12. (end)
Define the infinite square array m(n,k) by m(n,k) = (nk)^2 if n>=k>=0 and by m(n,k) = (k+n)*(kn) if 0<=n<=k. This contains A120070 below the diagonal. Then a(n) = sum_{k=0..n} m(n,k) + sum_{r=0..n} m(r,n), the "hook sum" of the terms to the left of m(n,n) and above m(n,n) with irrelevant (vanishing) terms on the diagonal.  J. M. Bergot, Aug 16 2013
a(n) = A245334(n+1,2), n > 0.  Reinhard Zumkeller, Aug 31 2014
a(n) is the sum of all pairs with repetition drawn from the set of odd numbers 2*n3. This is similar to A027480 but using the odd integers instead. Example using n=3 gives the odd numbers 1,3,5: 1+1, 1+3, 1+5, 3+3, 3+5,5+5 having a total of 36=a(3).  J. M. Bergot, Apr 05 2016
a(n) is the first Zagreb index of the complete graph K[n+1]. The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices. Alternately, it is the sum of the degree sums d(i)+d(j) over all edges ij of the graph.  Emeric Deutsch, Nov 07 2016
