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A000938
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Number of collinear point-triples in an n X n grid.
(Formerly M4527 N1919)
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44
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0, 8, 44, 152, 372, 824, 1544, 2712, 4448, 6992, 10332, 15072, 21012, 28688, 38520, 50880, 65480, 83640, 104676, 130264, 160556, 195848, 235600, 282840, 336384, 397136, 465876, 544464, 630684
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OFFSET
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2,2
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COMMENTS
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This is the no-3-in-line problem on an n X n grid.
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REFERENCES
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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.
R. K. Guy, Unsolved combinatorial problems, pp. 121-127 of D. J. A. Welsh, editor, Combinatorial Mathematics and Its Applications. Academic Press, NY, 1971.
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.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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R. H. Hardin, Table of n, a(n) for n=2..59, May 22 2010
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FORMULA
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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. [From Ignacio Larrosa Canestro (ilarrosa(AT)mundo-r.com), May 23 2010]
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EXAMPLE
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a(3) = 8: the 3 rows, 3 columns and 2 diagonals of a 3 X 3 grid.
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MAPLE
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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;
seq(a(n), n=2..30); # _Dennis Walsh_, Mar 02 2013
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MATHEMATICA
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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 *)
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CROSSREFS
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Cf. A000769. [From R. J. Mathar, May 21 2010]
Cf. A157882 for the 3-D version.
A000938 = Comb(n^2, 3) - A045996(n) [From Ignacio Larrosa Canestro (ilarrosa(AT)mundo-r.com), May 23 2010]
Sequence in context: A188148 A100583 A036464 * A165618 A059596 A181358
Adjacent sequences: A000935 A000936 A000937 * A000939 A000940 A000941
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KEYWORD
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nonn,nice
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Terms a(11) through a(30) from John W. Layman, Sep 21 2000.
Typo in formula corrected by David Bevan, Jan 09 2012
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STATUS
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approved
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