A000938
Number of collinear point-triples in an n X n grid.
Original entry on oeis.org
0, 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, 729744, 837744, 958384, 1091904, 1238520, 1400140, 1581384, 1776084
Offset: 1
a(3) = 8: the 3 rows, 3 columns and 2 diagonals of a 3 X 3 grid.
- 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).
- Alois P. Heinz, Table of n, a(n) for n = 1..1000 (Terms n=2..59 from R. H. Hardin)
- R. K. Guy and P. A. Kelly, The No-Three-Line Problem, Research Paper 33, Department of Mathematics, Univ. of Calgary, Calgary, Alberta, 1968. [Annotated scanned copy]
- R. K. Guy and P. A. Kelly, The No-Three-Line Problem, condensed version in Canad. Math. Bull. Vol. 11, pp. 527-531, 1968. [Annotated scanned copy]
- R. K. Guy, P. A. Kelly, N. J. A. Sloane, Correspondence, 1968-1971
This is the main diagonal of the array in
A334704.
-
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
-
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 *)
A178256
Number of ways to choose four collinear points from an n X n grid.
Original entry on oeis.org
0, 0, 0, 10, 64, 234, 660, 1524, 3156, 5928, 10428, 17154, 27340, 41506, 61176, 87756, 123216, 168420, 227208, 300054, 391920, 504886, 642604, 806424, 1006404, 1242024, 1519980, 1845150, 2226804, 2663574, 3175048, 3754936, 4420440, 5175840, 6030840
Offset: 1
R. H. Hardin, suggested by R. J. Mathar in the Sequence Fans Mailing List, May 24 2010
a(1) = a(2) = a(3) = 0 since there are no collinear point quadruples
a(4) = 4 rows + 4 columns + 2 diagonals = 10
a(5) = binomial(5,4)*(5 rows + 5 columns + 2 diagonals) + 4 secondary diagonals = 64
a(6) = binomial(6,4)*(6 rows + 6 columns + 2 diagonals) + binomial(5,4)*(4 secondary diagonals) + 4 third diagonals = 234
This is the main diagonal of
A334708.
A178257
Number of collinear point 5-tuples in an n X n cubical grid.
Original entry on oeis.org
0, 0, 0, 0, 12, 88, 364, 1120, 2876, 6432, 13116, 24640, 43800, 73656, 119160, 185616, 281064, 413008, 594888, 836784, 1157812, 1573912, 2109844, 2784608, 3638980, 4693888, 5996612, 7583488, 9514336, 11822904, 14603264, 17886272, 21778080
Offset: 1
R. H. Hardin, suggested by R. J. Mathar in the Sequence Fans Mailing List, May 24 2010
Three collinear points in an n X n grid
A000938 ; n X n X n grid
A157882
A178258
Number of collinear point 6-tuples in an n X n cubical grid.
Original entry on oeis.org
0, 0, 0, 0, 0, 14, 116, 536, 1824, 5100, 12432, 27248, 55148, 104282, 187120, 320244, 527976, 840276, 1300944, 1959780, 2888920, 4168470, 5910540, 8233080, 11310640, 15309580, 20477040, 27057280, 35393500, 45803810, 58784224, 74742016, 94324896
Offset: 1
R. H. Hardin, suggested by R. J. Mathar in the Sequence Fans Mailing List, May 24 2010
Three collinear points in an n X n grid
A000938 ; n X n X n grid
A157882
A178259
Number of collinear point 7-tuples in an n X n cubical grid.
Original entry on oeis.org
0, 0, 0, 0, 0, 0, 16, 148, 756, 2820, 8580, 22572, 53224, 115036, 231960, 441032, 799064, 1387368, 2324744, 3771176, 5952528, 9159392, 13791008, 20340036, 29476508, 41992820, 58953420, 81591788, 111535064, 150611100, 201257608
Offset: 1
R. H. Hardin, suggested by R. J. Mathar in the Sequence Fans Mailing List, May 24 2010
Three collinear points in an n X n grid
A000938 ; n X n X n grid
A157882
A178262
Number of collinear point 4-tuples in an n X n X n cubical grid.
Original entry on oeis.org
0, 0, 0, 76, 629, 2820, 9767, 26272, 63162, 135504, 269334, 489204, 870943, 1447460, 2319069, 3613088, 5496580, 8046288, 11646108, 16349868, 22751289, 31151020, 41928083, 55298280, 72845822, 94565584, 121269906, 154010268, 194468499
Offset: 1
R. H. Hardin, suggested by R. J. Mathar in the Sequence Fans Mailing List, May 24 2010
Three collinear points in an n X n grid
A000938 ; n X n X n grid
A157882
A178263
Number of collinear point 5-tuples in an n X n X n cubical grid.
Original entry on oeis.org
0, 0, 0, 0, 109, 984, 4833, 17216, 50778, 127440, 290514, 599904, 1172679, 2138920, 3744147, 6268512, 10184228, 15922368, 24423156, 36328992, 53149041, 76127736, 107316085, 148241856, 203111358, 273647888, 364606470, 479946816, 626565195
Offset: 1
R. H. Hardin, suggested by R. J. Mathar in the Sequence Fans Mailing List, May 24 2010
Three collinear points in an n X n grid
A000938 ; n X n X n grid
A157882
A178268
Number of collinear point triples in an n X n .. X n 4-dimensional cubical grid.
Original entry on oeis.org
0, 0, 272, 2960, 21680, 85584, 330032, 916832, 2402544, 5543936, 12188000, 23612496, 46235520, 82329552, 142808864, 239006304, 393144576, 612427040, 953341296, 1420731120, 2107948720, 3060955376, 4384446800, 6098267968
Offset: 1
R. H. Hardin, suggested by R. J. Mathar in the Sequence Fans Mailing List, May 24 2010
Three collinear points in an n X n grid
A000938 ; n X n X n grid
A157882
A178269
Number of collinear point 4-tuples in an n X n .. X n 4-dimensional cubical grid.
Original entry on oeis.org
0, 0, 0, 520, 5632, 31176, 135240, 429360, 1211208, 3028224, 6888216, 13961448, 28262776, 52206088, 91745904, 157574640, 263360352, 418048464, 657072528, 991543416, 1487807520, 2194825240, 3156000184, 4416200736, 6212710248
Offset: 1
R. H. Hardin, suggested by R. J. Mathar in the Sequence Fans Mailing List, May 24 2010
Three collinear points in an n X n grid
A000938 ; n X n X n grid
A157882
A178271
Number of collinear point 6-tuples in an n X n .. X n 4-dimensional cubical grid.
Original entry on oeis.org
0, 0, 0, 0, 0, 1400, 16520, 103040, 455040, 1602000, 4840032, 12815552, 31069112, 68954312, 144449440, 284461680, 537258960, 966579120
Offset: 1
R. H. Hardin, suggested by R. J. Mathar in the Sequence Fans Mailing List, May 24 2010
Three collinear points in an n X n grid
A000938 ; n X n X n grid
A157882
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