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A011379 a(n) = n^2*(n+1). 44

%I

%S 0,2,12,36,80,150,252,392,576,810,1100,1452,1872,2366,2940,3600,4352,

%T 5202,6156,7220,8400,9702,11132,12696,14400,16250,18252,20412,22736,

%U 25230,27900,30752,33792,37026,40460,44100,47952,52022,56316,60840

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

%C (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

%C 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_, Jan 15 2003

%C 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 (bhmd95(AT)yahoo.fr), May 09 2006

%C sum_{n>0} 1/a(n) = (Pi^2 - 6)/6 = 0.6449340.. [Jolley eq 272] - _Gary W. Adamson_, Dec 22 2006

%C a(2d-1) is the number of self-avoiding walk of length 3 in the d-dimensional hypercubic lattice. - _Michael Somos_, Sep 06 2006

%C a(n) mod 10 is periodic 5: repeat [0, 2, 2, 6, 0]. [Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Sep 05 2009]

%C 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*(d*k+d-2)/2, k=0..n-1) = n*(n+d)*(2*d*n+d-3)/6. - _Bruno Berselli_, Nov 18 2010

%C 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]

%C a(n+1) = sum of n-th row of the triangle in A195437. - _Reinhard Zumkeller_, Nov 23 2011

%C Partial sums of A049450. - _Omar E. Pol_, Jan 12 2013

%C From _Jon Perry_, May 11 2013: (Start)

%C Define a 'stable brick triangle' as:

%C -----

%C | c |

%C ---------

%C | a | | b |

%C ----------

%C 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.

%C 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.

%C So for i=j=n=2 we have 4:

%C 1 2 3 4

%C 2 2 2 2 2 2 2 2

%C 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)

%C 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

%C a(n) = A245334(n+1,2), n > 0. - _Reinhard Zumkeller_, Aug 31 2014

%C 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

%C 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

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

%D I. Gutman and K. C. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50, 2004, 83-92.

%H Vincenzo Librandi, <a href="/A011379/b011379.txt">Table of n, a(n) for n = 0..1000</a>

%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas for Some Functions on Finite Sets</a>

%H B. Berselli, A description of the recursive method in Comments lines: website <a href="http://www.lanostra-matematica.org/2008/12/sequenze-numeriche-e-procedimenti.html">Matem@ticamente</a> (in Italian).

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

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

%F a(n) = sum(sum((i+j), i=1..n), j=1..n), row sums of A126890 skipping numbers in the first column. - _Alexander Adamchuk_, Oct 12 2004

%F a(n) = 2*n*binomial(n+1,2) = 2*n*A000217(n). - _Arkadiusz Wesolowski_, Feb 10 2012

%F G.f.: 2*x*(1 + 2*x)/(1 - x)^4. - _Arkadiusz Wesolowski_, Feb 11 2012

%F a(n) = A000330(n) + A002412(n) = A000292(n) + A002413(n). - _Omar E. Pol_, Jan 11 2013

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

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

%t lst={}; Do[AppendTo[lst, n^3+n^2], {n,0,100}]; lst (* _Vladimir Joseph Stephan Orlovsky_, Jan 03 2009 *)

%t LinearRecurrence[{4,-6,4,-1},{0,2,12,36},40] (* _Harvey P. Dale_, Sep 13 2018 *)

%o (MAGMA) [n^2+n^3: n in [0..40]]; // _Vincenzo Librandi_, May 02 2011

%o (Haskell)

%o a011379 n = a000290 n + a000578 n -- _Reinhard Zumkeller_, Apr 28 2013

%o (PARI) a(n)=n^3+n^2 \\ _Charles R Greathouse IV_, Apr 06 2016

%Y Cf. A000290, A000578, A005449, A022549, A045991, A245334.

%K nonn,easy

%O 0,2

%A Glen Burch (gburch(AT)erols.com); _Felice Russo_

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Last modified January 17 06:55 EST 2019. Contains 319207 sequences. (Running on oeis4.)