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%I #83 Jan 20 2024 17:55:40
%S 0,0,2,12,40,100,210,392,672,1080,1650,2420,3432,4732,6370,8400,10880,
%T 13872,17442,21660,26600,32340,38962,46552,55200,65000,76050,88452,
%U 102312,117740,134850,153760,174592,197472,222530,249900,279720,312132
%N a(n) = n^2*(n^2 - 1)/6.
%C Number of equilateral triangles in rhombic portion of side n+1 in hexagonal lattice.
%C The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.
%C Sum of squared distances on n X n board between n queens each on its own row and column. - _Zak Seidov_, Sep 04 2002
%C For queens "each on its column and row" the sum of squared distances does not depend on configuration - while sum of distances does.
%C Number of cycles of length 3 in the bishop's graph associated with an (n+1) X (n+1) chessboard. - Anton Voropaev (anton.n.voropaev(AT)gmail.com), Feb 01 2009
%C a(n) is number of ways to place 3 queens on an (n+1) X (n+1) chessboard so that they diagonally attack each other exactly 3 times. The maximal possible attack number, p=binomial(k,2)=3 for k=3 queens, is achievable only when all queens are on the same diagonal. In graph-theory representation they thus form the corresponding complete graph. - _Antal Pinter_, Dec 27 2015
%C From a(1), convolution of the oblong numbers (A002378) with the odd numbers (A005408). - _Bruno Berselli_, Oct 24 2016
%C Consider the partitions of 2n into two parts (p,q) where p <= q. Then a(n) is the total volume of the family of rectangular prisms with dimensions p, p and |q-p|. - _Wesley Ivan Hurt_, Apr 15 2018
%D James Propp, Enumeration of matchings: problems and progress, pp. 255-291 in L. J. Billera et al., eds, New Perspectives in Algebraic Combinatorics, Cambridge, 1999 (see Problem 6).
%H Vincenzo Librandi, <a href="/A008911/b008911.txt">Table of n, a(n) for n = 0..10000</a>
%H Gabriele Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A2.html">Home page for hexagonal (or triangular) lattice A2</a>
%H James Propp, Enumeration of matchings: problems and progress, in L. J. Billera et al. (eds.), <a href="http://www.msri.org/publications/books/Book38/contents.html">New Perspectives in Algebraic Combinatorics</a>, 1999.
%H James Propp, <a href="http://faculty.uml.edu/jpropp/update.pdf">Updated article</a>, 2009.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F G.f.: 2*x^2*(1+x)/(1-x)^5.
%F a(n) = 2*A002415(n) = A047928(n-1)/6 = A083374(n-1)/3 = A006011(n)*2/3. - _Zerinvary Lajos_, May 09 2007
%F a(n) = n*binomial(n+1,3). - _Martin Renner_, Apr 03 2011
%F a(n+1) = (n+1)*A000292(n). - _Tom Copeland_, Sep 13 2011
%F From _G. C. Greubel_, Sep 13 2019: (Start)
%F a(n) = binomial(n^2,2)/3.
%F E.g.f.: x^2*(6 + 6*x + x^2)*exp(x)/6. (End)
%F From _Amiram Eldar_, Nov 02 2021: (Start)
%F Sum_{n>=2} 1/a(n) = 21/2 - Pi^2.
%F Sum_{n>=2} (-1)^n/a(n) = (Pi^2 - 9)/2. (End)
%F a(n) = Sum_{j=0..n-1} binomial(n,2) + binomial(n,3). - _Detlef Meya_, Jan 20 2024
%e a(2)=2 because on 2 X 2 board queens "each on its column and row" may take only two angular cells, then squared distance is 1^2+1^2=2. a(3)=12 because on 3 X 3 board queens "each on its column and row" make only two essentially distinct configurations: {1,2,3}, {1,3,2} and in both cases the sum of three squared distances is 12.
%e G.f.: 2*x^2 + 12*x^3 + 40*x^4 + 100*x^5 + 210*x^6 + 392*x^7 + 672*x^8 + ...
%p A008911 := n->n^2*(n^2-1)/6; seq(A008911(n), n=0..40);
%t a[m_]:= m^2(m^2-1)/6;
%t Binomial[Range[0,40]^2, 2]/3 (* _G. C. Greubel_, Sep 13 2019 *)
%o (PARI) {a(n) = n^2*(n^2-1)/6};
%o (Magma) [n^2*(n^2-1)/6: n in [0..40]]; // _Vincenzo Librandi_, Sep 14 2011
%o (Sage) [n^2*(n^2-1)/6 for n in (0..40)] # _G. C. Greubel_, Sep 13 2019
%o (GAP) List([0..40], n-> n^2*(n^2-1)/6); # _G. C. Greubel_, Sep 13 2019
%Y Cf. A002415, A006011, A047928, A083374.
%Y Cf. A002378, A005408.
%Y Convolution of the oblong numbers with the even numbers: A033488.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_, _R. K. Guy_
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