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A011379 a(n) = n^2*(n+1). 51
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 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, 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..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, Jan 15 2003

Y values of solutions of the equation: (X-Y)^3-X*Y=0. X values are a(n)=n*(n+1)^2 (see A045991) - Mohamed Bouhamida, 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

a(n) mod 10 is periodic 5: repeat [0, 2, 2, 6, 0]. - Mohamed Bouhamida, Sep 05 2009

This sequence is related to A005449 by a(n) = n*A005449(n)-sum(A005449(i), i=0..n-1), and this is the case d=3 in the identity n^2*(d*n+d-2)/2 - Sum_{k=0..n-1} k*(d*k+d-2)/2 = n*(n+d)*(2*d*n+d-3)/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) is the sum of n-th 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) = (n-k)^2 if n >= k >= 0 and by m(n,k) = (k+n)*(k-n) 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*n-3. 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

REFERENCES

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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Ivan Gutman and Kinkar Ch. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50, 2004, 83-92.

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

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

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

FORMULA

a(n) = 2*A002411(n).

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

a(n) = 2*n*binomial(n+1,2) = 2*n*A000217(n). - Arkadiusz Wesolowski, Feb 10 2012

G.f.: 2*x*(1 + 2*x)/(1 - x)^4. - Arkadiusz Wesolowski, Feb 11 2012

a(n) = A000330(n) + A002412(n) = A000292(n) + A002413(n). - Omar E. Pol, Jan 11 2013

a(n) = A245334(n+1,2), n > 0. - Reinhard Zumkeller, Aug 31 2014

Sum_{n>=1) 1/a(n) = -(A013661-1). - R. J. Mathar, Oct 18 2019

Sum_{n>=1} (-1)^(n+1)/a(n) = 1 + Pi^2/12 - 2*log(2). - Amiram Eldar, Jul 04 2020

E.g.f.: exp(x)*x*(2 + 4*x + x^2). - Stefano Spezia, May 20 2021

EXAMPLE

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

MAPLE

A011379:=n->n^2*(n+1); seq(A011379(n), n=0..40); # Wesley Ivan Hurt, Feb 25 2014

MATHEMATICA

Table[n^3+n^2, {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Jan 03 2009, modified by G. C. Greubel, Aug 10 2019 *)

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

PROG

(Magma) [n^2+n^3: n in [0..40]]; // Vincenzo Librandi, May 02 2011

(Haskell)

a011379 n = a000290 n + a000578 n -- Reinhard Zumkeller, Apr 28 2013

(PARI) a(n)=n^3+n^2 \\ Charles R Greathouse IV, Apr 06 2016

(Sage) [n^2*(n+1) for n in (0..40)] # G. C. Greubel, Aug 10 2019

(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.

Sequence in context: A246426 A353503 A176583 * A338610 A073404 A141208

Adjacent sequences: A011376 A011377 A011378 * A011380 A011381 A011382

KEYWORD

nonn,easy

AUTHOR

Glen Burch (gburch(AT)erols.com), Felice Russo

STATUS

approved

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Last modified November 26 21:15 EST 2022. Contains 358362 sequences. (Running on oeis4.)