%I M4527 N1919 #57 Jun 19 2020 23:50:05
%S 0,0,8,44,152,372,824,1544,2712,4448,6992,10332,15072,21012,28688,
%T 38520,50880,65480,83640,104676,130264,160556,195848,235600,282840,
%U 336384,397136,465876,544464,630684,729744,837744,958384,1091904,1238520,1400140,1581384,1776084
%N Number of collinear point-triples in an n X n grid.
%C This is related to the no-3-in-line problem on an n X n grid.
%D M. A. Adena, D. A. Holton and P. A. Kelly, Some thoughts on the no-three-in-line problem, pp. 6-17 of Combinatorial Mathematics (Proceedings 2nd Australian Conf.), Lect. Notes Math. 403, 1974.
%D R. K. Guy, Unsolved combinatorial problems, pp. 121-127 of D. J. A. Welsh, editor, Combinatorial Mathematics and Its Applications. Academic Press, NY, 1971.
%D R. K. Guy and P. A. Kelly, The No-Three-Line Problem. Research Paper 33, Department of Mathematics, Univ. of Calgary, Calgary, Alberta, 1968. Condensed version in Canad. Math. Bull. Vol. 11, pp. 527-531, 1968.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Alois P. Heinz, <a href="/A000938/b000938.txt">Table of n, a(n) for n = 1..1000</a> (Terms n=2..59 from R. H. Hardin)
%H R. K. Guy and P. A. Kelly, <a href="/A000755/a000755_1.pdf">The No-Three-Line Problem</a>, Research Paper 33, Department of Mathematics, Univ. of Calgary, Calgary, Alberta, 1968. [Annotated scanned copy]
%H R. K. Guy and P. A. Kelly, <a href="/A000755/a000755_2.pdf">The No-Three-Line Problem</a>, condensed version in Canad. Math. Bull. Vol. 11, pp. 527-531, 1968. [Annotated scanned copy]
%H R. K. Guy, P. A. Kelly, N. J. A. Sloane, <a href="/A000755/a000755.pdf">Correspondence, 1968-1971</a>
%F a(n) = 2*Sum(Sum((n - k + 1)*(n - m + 1)*gcd(k - 1, m - 1), k, 2, n), m, 2, n) - n^2(n^2 - 1)/6. - _Ignacio Larrosa Cañestro_, May 23 2010
%F a(n) = binomial(n^2, 3) - A045996(n). - _Ignacio Larrosa Cañestro_, May 23 2010
%e a(3) = 8: the 3 rows, 3 columns and 2 diagonals of a 3 X 3 grid.
%p a:=n->2*sum(sum((n - k + 1)*(n - m + 1)*igcd(k - 1, m - 1), k= 2.. n), m= 2.. n) - n^2*(n^2 - 1)/6;
%p seq(a(n),n=2..30); # _Dennis P. Walsh_, Mar 02 2013
%t a[n_] := 2*Sum[(n - k + 1)*(n - m + 1)*GCD[k - 1, m - 1], {m, 2, n}, {k, 2, n}] - n^2*((n^2 - 1)/6); Table[a[n], {n, 2, 30}] (* _Jean-François Alcover_, Jul 11 2012, after _Ignacio Larrosa Cañestro_ *)
%Y This is the main diagonal of the array in A334704.
%Y Cf. A000769, A272651.
%Y Cf. A157882 for the 3-D version.
%K nonn,nice
%O 1,3
%A _N. J. A. Sloane_
%E Terms a(11) through a(30) from _John W. Layman_, Sep 21 2000
%E Typo in formula corrected by _David Bevan_, Jan 09 2012
%E Offset changed to 1 and initial 0 added. - _N. J. A. Sloane_, Jun 19 2020
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