

A000938


Number of collinear pointtriples in an n X n grid.
(Formerly M4527 N1919)


46



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

2,2


COMMENTS

This is the no3inline problem on an n X n grid.


REFERENCES

M. A. Adena, D. A. Holton and P. A. Kelly, Some thoughts on the nothreeinline problem, pp. 617 of Combinatorial Mathematics (Proceedings 2nd Australian Conf.), Lect. Notes Math. 403, 1974.
R. K. Guy, Unsolved combinatorial problems, pp. 121127 of D. J. A. Welsh, editor, Combinatorial Mathematics and Its Applications. Academic Press, NY, 1971.
R. K. Guy and P. A. Kelly, The NoThreeLine Problem. Research Paper 33, Department of Mathematics, Univ. of Calgary, Calgary, Alberta, 1968. Condensed version in Canad. Math. Bull. Vol. 11, pp. 527531, 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).


LINKS

R. H. Hardin, Table of n, a(n) for n=2..59, May 22 2010


FORMULA

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


EXAMPLE

a(3) = 8: the 3 rows, 3 columns and 2 diagonals of a 3 X 3 grid.


MAPLE

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 P. Walsh, Mar 02 2013


MATHEMATICA

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}] (* JeanFrançois Alcover, Jul 11 2012, after Ignacio Larrosa Cañestro *)


CROSSREFS

Cf. A000769. [From R. J. Mathar, May 21 2010]
Cf. A157882 for the 3D version.
A000938 = Comb(n^2, 3)  A045996(n).  Ignacio Larrosa Cañestro, May 23 2010
Sequence in context: A188148 A100583 A036464 * A165618 A250285 A059596
Adjacent sequences: A000935 A000936 A000937 * A000939 A000940 A000941


KEYWORD

nonn,nice


AUTHOR

N. J. A. Sloane.


EXTENSIONS

Terms a(11) through a(30) from John W. Layman, Sep 21 2000.
Typo in formula corrected by David Bevan, Jan 09 2012


STATUS

approved



