%I #195 Aug 04 2024 02:54:28
%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) = (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
%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_, May 09 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_, 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=0..n-1} k*(d*k+d-2)/2 = 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) is the 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) 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
%C a(n-2) 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
%D L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, pp. 50, 64.
%H Vincenzo Librandi, <a href="/A011379/b011379.txt">Table of n, a(n) for n = 0..1000</a>
%H Bruno 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 Allan Bickle and Zhongyuan Che, <a href="https://doi.org/10.1016/j.dam.2023.01.020">Irregularities of Maximal k-degenerate Graphs</a>, Discrete Applied Math. 331 (2023) 70-87.
%H Allan Bickle, <a href="https://doi.org/10.20429/tag.2024.000105">A Survey of Maximal k-degenerate Graphs and k-Trees</a>, Theory and Applications of Graphs 0 1 (2024) Article 5.
%H Ivan Gutman and Kinkar Ch. Das, <a href="http://match.pmf.kg.ac.rs/electronic_versions/Match50/match50_83-92.pdf">The first Zagreb index 30 years after</a>, MATCH Commun. Math. Comput. Chem. 50, 2004, 83-92.
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas for Some Functions on Finite Sets</a>
%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_{j=1..n} (Sum_{i=1..n} (i+j)), row sums of A126890 skipping numbers in the first column. - _Alexander Adamchuk_, Oct 12 2004
%F Sum_{n>0} 1/a(n) = (Pi^2 - 6)/6 = 0.6449340... [Jolley eq 272] - _Gary W. Adamson_, Dec 22 2006
%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
%F a(n) = A245334(n+1,2), n > 0. - _Reinhard Zumkeller_, Aug 31 2014
%F Sum_{n>=1} 1/a(n) = A013661-1. - _R. J. Mathar_, Oct 18 2019 [corrected by _Jason Yuen_, Aug 04 2024]
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 1 + Pi^2/12 - 2*log(2). - _Amiram Eldar_, Jul 04 2020
%F E.g.f.: exp(x)*x*(2 + 4*x + x^2). - _Stefano Spezia_, May 20 2021
%F a(n) = n*A002378(n) = A000578(n) + A000290(n). - _J.S. Seneschal_, Jun 18 2024
%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 Table[n^3+n^2, {n,0,40}] (* _Vladimir Joseph Stephan Orlovsky_, Jan 03 2009, modified by _G. C. Greubel_, Aug 10 2019 *)
%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
%o (Sage) [n^2*(n+1) for n in (0..40)] # _G. C. Greubel_, Aug 10 2019
%o (GAP) List([0..40], n-> n^2*(n+1) ); # _G. C. Greubel_, Aug 10 2019
%Y Cf. A000217, A000290, A000292, A000330, A000578, A002378, A002411, A002412, A002413, A005449, A013661, A022549, A027480, A045991, A049450, A120070, A126890, A195437, A245334.
%Y Cf. A011379, A181617, A270205 (sigma irregularities of maximal k-degenerate graphs).
%K nonn,easy
%O 0,2
%A Glen Burch (gburch(AT)erols.com), _Felice Russo_