

A011379


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


61



0, 2, 12, 36, 80, 150, 252, 392, 576, 810, 1100, 1452, 1872, 2366, 2940, 3600, 4352, 5202, 6156, 7220, 8400, 9702, 11132, 12696, 14400, 16250, 18252, 20412, 22736, 25230, 27900, 30752, 33792, 37026, 40460, 44100, 47952, 52022, 56316, 60840
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,2


COMMENTS

(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) = (Product_{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, May 09 2006
a(2d1) is the number of selfavoiding walk of length 3 in the ddimensional hypercubic lattice.  Michael Somos, Sep 06 2006
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)/2  Sum_{k=0..n1} k*(d*k+d2)/2 = 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
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) 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
a(n2) is the maximum sigma irregularity over all trees with n vertices. The extremal graphs are stars. (The sigma irregularity of a graph is the sum of squares of the differences between the degrees over all edges of the graph.)  Allan Bickle, Jun 14 2023


REFERENCES

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


LINKS

Bruno Berselli, A description of the recursive method in Comments lines: website Matem@ticamente (in Italian).


FORMULA

a(n) = Sum_{j=1..n} (Sum_{i=1..n} (i+j)), row sums of A126890 skipping numbers in the first column.  Alexander Adamchuk, Oct 12 2004
Sum_{n>0} 1/a(n) = (Pi^2  6)/6 = 0.6449340... [Jolley eq 272]  Gary W. Adamson, Dec 22 2006
Sum_{n>=1} (1)^(n+1)/a(n) = 1 + Pi^2/12  2*log(2).  Amiram Eldar, Jul 04 2020


EXAMPLE

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


MAPLE



MATHEMATICA

LinearRecurrence[{4, 6, 4, 1}, {0, 2, 12, 36}, 40] (* Harvey P. Dale, Sep 13 2018 *)


PROG

(Haskell)
(GAP) List([0..40], n> n^2*(n+1) ); # G. C. Greubel, Aug 10 2019


CROSSREFS

Cf. A000217, A000290, A000292, A000330, A000578, A002411, A002412, A002413, A005449, A013661, A022549, A027480, A045991, A049450, A120070, A126890, A195437, A245334.


KEYWORD

nonn,easy


AUTHOR



STATUS

approved



