A000938 -id:A000938 - cp's OEIS frontend

A157882 Number of collinear point-triples in the n X n X n cube.

0, 0, 49, 376, 1858, 5696, 16427, 36992, 78204, 150672, 277005, 463624, 776494, 1212208, 1845911, 2749568, 4023608, 5654976, 7915497, 10730616, 14487706, 19290352, 25343011, 32580752, 41959412, 53240624, 66913605, 83330712

Offset: 1

R. J. Mathar, May 21 2010

nonn

A 3D variant of A000938.

			For n=3, for example, the 49 collinear triples have coordinates (sorting according to the base-n representation of numbers from 0 to n^3-1):
[0, 0, 0], [1, 0, 0], [2, 0, 0]
[0, 0, 0], [0, 1, 0], [0, 2, 0]
[0, 0, 0], [1, 1, 0], [2, 2, 0]
[0, 0, 0], [0, 0, 1], [0, 0, 2]
[0, 0, 0], [1, 0, 1], [2, 0, 2]
[0, 0, 0], [0, 1, 1], [0, 2, 2]
[0, 0, 0], [1, 1, 1], [2, 2, 2]
[1, 0, 0], [1, 1, 0], [1, 2, 0]
[1, 0, 0], [1, 0, 1], [1, 0, 2]
[1, 0, 0], [1, 1, 1], [1, 2, 2]
[2, 0, 0], [1, 1, 0], [0, 2, 0]
[2, 0, 0], [2, 1, 0], [2, 2, 0]
[2, 0, 0], [1, 0, 1], [0, 0, 2]
[2, 0, 0], [2, 0, 1], [2, 0, 2]
[2, 0, 0], [1, 1, 1], [0, 2, 2]
[2, 0, 0], [2, 1, 1], [2, 2, 2]
[0, 1, 0], [1, 1, 0], [2, 1, 0]
[0, 1, 0], [0, 1, 1], [0, 1, 2]
[0, 1, 0], [1, 1, 1], [2, 1, 2]
[1, 1, 0], [1, 1, 1], [1, 1, 2]
[2, 1, 0], [1, 1, 1], [0, 1, 2]
[2, 1, 0], [2, 1, 1], [2, 1, 2]
[0, 2, 0], [1, 2, 0], [2, 2, 0]
[0, 2, 0], [0, 1, 1], [0, 0, 2]
[0, 2, 0], [1, 1, 1], [2, 0, 2]
[0, 2, 0], [0, 2, 1], [0, 2, 2]
[0, 2, 0], [1, 2, 1], [2, 2, 2]
[1, 2, 0], [1, 1, 1], [1, 0, 2]
[1, 2, 0], [1, 2, 1], [1, 2, 2]
[2, 2, 0], [1, 1, 1], [0, 0, 2]
[2, 2, 0], [2, 1, 1], [2, 0, 2]
[2, 2, 0], [1, 2, 1], [0, 2, 2]
[2, 2, 0], [2, 2, 1], [2, 2, 2]
[0, 0, 1], [1, 0, 1], [2, 0, 1]
[0, 0, 1], [0, 1, 1], [0, 2, 1]
[0, 0, 1], [1, 1, 1], [2, 2, 1]
[1, 0, 1], [1, 1, 1], [1, 2, 1]
[2, 0, 1], [1, 1, 1], [0, 2, 1]
[2, 0, 1], [2, 1, 1], [2, 2, 1]
[0, 1, 1], [1, 1, 1], [2, 1, 1]
[0, 2, 1], [1, 2, 1], [2, 2, 1]
[0, 0, 2], [1, 0, 2], [2, 0, 2]
[0, 0, 2], [0, 1, 2], [0, 2, 2]
[0, 0, 2], [1, 1, 2], [2, 2, 2]
[1, 0, 2], [1, 1, 2], [1, 2, 2]
[2, 0, 2], [1, 1, 2], [0, 2, 2]
[2, 0, 2], [2, 1, 2], [2, 2, 2]
[0, 1, 2], [1, 1, 2], [2, 1, 2]
[0, 2, 2], [1, 2, 2], [2, 2, 2]

R. H. Hardin, Table of n, a(n) for n = 1..60
Hanno Lefmann, No l Grid-Points in spaces of small dimension, Lecture Notes in Comp. Sci. 5034 (2008) 259-270 [Provides motivation]
Attilo Por and David R. Wood, No-Three-in-Line-in-3D, Algorithmica 47 (4) (2007) 481-488, [Provides motivation]
Wikipedia, No-three-in-line problem.

Cf. A178256-A178298, A178351-A178353.

# return true if xtrip1, xtrip2 and xtrip3 are three collinear points in 3D
iscolin := proc(xtrip1,xtrip2,xtrip3)
local diff21x, diff21y, diff21z, diff31x, diff31y, diff31z ;
# build the difference vectors diff2=xtrip2-xtrip1 and diff3=xtrip3-xtrip1
# and test whether diff2=t*diff3 with some parameter t
diff21x := xtrip2[1]-xtrip1[1] ;
diff21y := xtrip2[2]-xtrip1[2] ;
diff21z := xtrip2[3]-xtrip1[3] ;
diff31x := xtrip3[1]-xtrip1[1] ;
diff31y := xtrip3[2]-xtrip1[2] ;
diff31z := xtrip3[3]-xtrip1[3] ;
if xtrip1 = xtrip2 or xtrip2 = xtrip3 or xtrip1 = xtrip3 then
error("degen triple") ;
end if ;
# is diff31[] = t * diff21[] ?
if diff21x = 0 then
if diff31x = 0 then
# both difference vectors in the y-z plane
if diff21y = 0 then
if diff31y = 0 then
# both diff vects on the z-axis
return true;
else
# one on the z-axis, the other not
return false;
end if;
else
if diff31y = 0 then
# one on the z-axis, the other one not
return false;
else
# general directions in the y-z plane
t := diff31y/diff21y ;
if t*diff21z = diff31z then
return true ;
else
return false;
end if;
end if;
end if;
else
# one diff vector in the y-z plane, the other not
return false;
end if;
else
if diff31x = 0 then
# one diff vector in the y-z plane, the other not
return false;
else
t := diff31x/diff21x ;
if t*diff21y = diff31y and t*diff21z = diff31z then
return true;
else
return false;
end if;
end if;
end if;
end proc:
# convert a number n=0,1,2,3,... into a triple [n1,n2,n3], all 0<=niA000938: the number of collinear triples in n X n X n
num3in := proc(n)
local a,ncub,nlin1,nlin2,xtrip2,xtrip3 ;
a := 0 ;
ncub := n^3 ;
# linearized index of first point
for nlin1 from 0 to ncub-1 do
xtrip1 := linidx(nlin1,n) ; # [x,y,z], 0<=x,y,z

Terms a(7) onwards from R. H. Hardin, May 21 2010

Replaced an invalid reference by Wikipedia and converted others to URL's R. J. Mathar, Jun 21 2010

A045996 Number of triangles in an n X n grid (or geoplane).

Original entry on oeis.org

0, 4, 76, 516, 2148, 6768, 17600, 40120, 82608, 157252, 280988, 477012, 775172, 1214768, 1844512, 2725000, 3930384, 5550844, 7692300, 10482124, 14066996, 18619128, 24337056, 31449200, 40212160, 50921316, 63907468, 79542108

Offset: 1

Ignacio Larrosa Cañestro

The triangles must have nonzero area -- their vertices must not be collinear.

The degenerate (i.e., collinear) triangles are counted in A000938. The 1000-term b-file there could be used to produce a 1000-term b-file for the present sequence. - N. J. A. Sloane, Jun 19 2020

			a(2)=4 because 4 isosceles right triangles can be placed on a 2 X 2 grid.

Vaclav Kotesovec, Table of n, a(n) for n = 1..1000
I. L. Canestro, Checkerboard, sci.math 22 Oct 2000 [broken link]
I. L. Canestro, Checkerboard, sci.math 22 Oct 2000 [Cached copy]

Cf. A000938.

a[n_] := ((n - 1)^2*n^2*(n + 1)^2)/6 - 2*Sum[(n - k + 1)*(n - l + 1)*GCD[k - 1, l - 1], {k, 2, n}, {l, 2, n}]; Array[a, 28] (* Robert G. Wilson v, May 23 2010 *)

a(n) = ((n-1)^2*n^2*(n+1)^2)/6 - 2*Sum_{m=2..n} Sum_{k=2..n} (n-k+1)*(n-m+1)*gcd(k-1, m-1).

a(n) = binomial(n^2,3) - A000938(n). - R. J. Mathar, May 21 2010

A000769 No-3-in-line problem: number of inequivalent ways of placing 2n points on an n X n grid so that no 3 are in a line.

Original entry on oeis.org

0, 1, 1, 4, 5, 11, 22, 57, 51, 156, 158, 566, 499, 1366, 3978, 5900, 7094, 19204

Offset: 1

N. J. A. Sloane

This means no three points on any line, not just lines in the X or Y directions.
A000755 gives the total number of solutions (as opposed to the number of equivalence classes).
It is conjectured that a(n)=0 for all sufficiently large n.
Flammenkamp's web site reports that at least one solution is known for all n <= 46 and n=48, 50, 52.
From R. K. Guy, Oct 22 2004: (Start)
I got the no-three-in-line problem from Heilbronn over 50 years ago. See Section F4 in UPINT.
In Canad. Math. Bull. 11 (1968) 527-531, MR 39 #129, Guy & Kelly conjecture that, for large n, at most (c + eps)*n points can be selected, where 3*c^3 = 2*Pi^2, i.e., c ~ 1.87.
As recently as last March, Gabor Ellmann pointed out an error in our heuristic reasoning, which, when corrected, gives 3*c^2 = Pi^2, or c ~ 1.813799. (End)

			a(3) = 1:
  X X o
  X o X
  o X X

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. B. Anderson, Journal of Combinatorial Theory Series A, V.27/1979 pp. 365 - 366.
D. Craggs and R. Hughes-Jones, Journal of Combinatorial Theory Series A, V. 20/1976 pp. 363-364.
H. E. Dudeney, Amusements in Mathematics, Nelson, Edinburgh 1917, pp. 94, 222.
M. Gardner, Scientific American V236 / March 1977, pp. 139-140.
M. Gardner, Penrose Tiles to Trapdoor Ciphers. Freeman, NY, 1989, p. 69.
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, Unsolved Problems Number Theory, Section F4.
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.
R. R. Hall, T. H. Jackson, A. Sudberry and K. Wild, Journal of Combinatorial Theory Series A, V.18/1975 pp. 336-341.
H. Harborth, P. Oertel and T. Prellberg, Discrete Math. V73/1988 pp. 89-90.
T. Kløve, Journal of Combinatorial Theory Series A, V.24/1978 pp. 126-127.
T. Kløve, Journal of Combinatorial Theory Series A, V.26/1979 pp. 82-83.
K. F. Roth, Journal London Math. Society V.26 / 1951, p. 204.
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).

Benjamin Chaffin, No-Three-In-Line Problem.
Alec S. Cooper, Oleg Pikhurko, John R. Schmitt and Gregory S. Warrington, Martin Gardner's minimum no-3-in-a-line problem, arXiv:1206.5350 [math.CO]. Also Amer. Math. Monthly, 121 (2014), 213-221.
Achim Flammenkamp, Progress in the no-three-in-line problem.
Achim Flammenkamp, Solutions of the no-three-in-line problem.
Achim Flammenkamp, Progress in the no-three-in-line problem, J. Combinat. Theory A 60 (1992), 305-311.
Achim Flammenkamp, Progress in the no-three-in-line problem. II, J. Combin. Theory Ser. A 81 (1998), no. 1, 108-113.
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.
S. V. Ullas Chandran, Sandi Klavžar, and James Tuite, The General Position Problem: A Survey, arXiv:2501.19385 [math.CO], 2025. See p. 4.
Eric Weisstein's World of Mathematics, Point Lattice.
Eric Weisstein's World of Mathematics, No-Three-in-a-Line-Problem.

Cf. A000755, A000938, A037185, A037186, A037187, A037188, A037189, A047840, A212807, A235453.
See A272651 for the maximal number of no-3-in-line points on an n X n grid, and A277433 for minimal saturated.
Cf. A194136 (triangular grid), A280537 (3D grid, no 4 in plane).

a(17) and a(18) from Benjamin Chaffin, Apr 05 2006
Minor edits from N. J. A. Sloane, May 25 2010
Edited by N. J. A. Sloane, Mar 19 2013 at the suggestion of Dominique Bernardi

A334704 Triangle read by rows: T(n,k) (1 <= k <= n) = number of ways to choose three collinear points from an n X k grid of points.

Original entry on oeis.org

0, 0, 0, 1, 2, 8, 4, 8, 20, 44, 10, 20, 43, 84, 152, 20, 40, 78, 140, 240, 372, 35, 70, 130, 224, 369, 558, 824, 56, 112, 200, 332, 528, 780, 1132, 1544, 84, 168, 293, 472, 734, 1064, 1519, 2052, 2712, 120, 240, 410, 648, 988, 1408, 1982, 2652, 3480, 4448

Offset: 1

N. J. A. Sloane, Jun 13 2020

It follows from the definitions that T(n,k) + A334705(n,k) = A334703(n,k) for 1 <= k <= n.

			Triangle begins:
0,
0, 0,
1, 2, 8,
4, 8, 20, 44,
10, 20, 43, 84, 152,
20, 40, 78, 140, 240, 372,
35, 70, 130, 224, 369, 558, 824,
56, 112, 200, 332, 528, 780, 1132, 1544,
84, 168, 293, 472, 734, 1064, 1519, 2052, 2712,
120, 240, 410, 648, 988, 1408, 1982, 2652, 3480, 4448,
165, 330, 556, 864, 1295, 1826, 2542, 3372, 4393, 5586, 6992,
...
This is the lower half of a symmetric array. The full symmetric array begins:
0, 0, 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, ...
0, 0, 2, 8, 20, 40, 70, 112, 168, 240, 330, 440, ...
1, 2, 8, 20, 43, 78, 130, 200, 293, 410, 556, 732, ...
4, 8, 20, 44, 84, 140, 224, 332, 472, 648, 864, 1120, ...
10, 20, 43, 84, 152, 240, 369, 528, 734, 988, 1295, 1652, ...
20, 40, 78, 140, 240, 372, 558, 780, 1064, 1408, 1826, 2304, ...
35, 70, 130, 224, 369, 558, 824, 1132, 1519, 1982, 2542, 3172, ...
56, 112, 200, 332, 528, 780, 1132, 1544, 2052, 2652, 3372, 4172, ...
84, 168, 293, 472, 734, 1064, 1519, 2052, 2712, 3480, 4393, 5396, ...
120, 240, 410, 648, 988, 1408, 1982, 2652, 3480, 4448, 5586, 6824, ...
165, 330, 556, 864, 1295, 1826, 2542, 3372, 4393, 5586, 6992, 8508, ...
220, 440, 732, 1120, 1652, 2304, 3172, 4172, 5396, 6824, 8508, 10332, ...
...

Tom Duff, Values of T(n,k) for grids of size up to 50 X 50

This is a companion to the triangles A334703 and A334705.

Rows (or columns) 1,2,3,4 of the full array are A000292, A007290, A057566, A334706. The main diagonal is A000938.

Rows 6 onwards from Tom Duff. - N. J. A. Sloane, Jun 19 2020

A235453 Triangle T(n, k) = Number of non-equivalent (mod D_4) ways to arrange k indistinguishable points on an n X n square grid so that no three of them are collinear. Triangle read by rows.

Original entry on oeis.org

1, 0, 1, 2, 1, 1, 3, 8, 13, 15, 5, 1, 3, 21, 70, 181, 217, 142, 28, 4, 6, 49, 290, 1253, 3192, 4699, 3385, 1076, 110, 5, 6, 93, 867, 6044, 27041, 77970, 134353, 129929, 62177, 12511, 717, 11, 10, 171, 2266, 22302, 149217, 672506, 1958674, 3531747, 3695848, 2068757

Offset: 1

Heinrich Ludwig, Jan 12 2014

The triangle T(n, k) is irregularly shaped: 1 <= k <= 2n. First row corresponds to n = 1.
Without the restriction "non-equivalent (mod D_4)" the numbers are given by triangle A194193. (But this one is read by antidiagonals!)
T(n, 2n) = A000769(n).
2n is an upper bound on the number of points that can be placed on the grid. For large n, it is conjectured that this bound is not reached (see MathWorld link).

			Triangle begins
1,  0;
1,  2,   1,    1;
3,  8,  13,   15,     5,     1;
3, 21,  70,  181,   217,   142,     28,      4;
6, 49, 290, 1253,  3192,  4699,   3385,   1076,   110,     5;
6, 93, 867, 6044, 27041, 77970, 134353, 129929, 62177, 12511, 717, 11;
...

Heinrich Ludwig, Table of n, a(n) for n = 1..99
Achim Flammenkamp, Progress in the no-three-in-line problem
Eric Weisstein's World of Mathematics, No-Three-in-a-Line-Problem

Cf. A194193, A000938, A000769.
Column 1 is A008805
Column 2 is A014409
Column 3 is A235454
Column 4 is A235455
Column 5 is A235456
Column 6 is A235457
Column 7 is A235458

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

nonn

			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

Tomas Rokicki and Tom Duff, Table of n, a(n) for n = 1..1000 (First 48 terms from R. H. Hardin.)

Cf. A000938, A157882.

This is the main diagonal of A334708.

A178208 Number of ways to choose three points in an (n X n)-grid (or geoplane).

Original entry on oeis.org

0, 4, 84, 560, 2300, 7140, 18424, 41664, 85320, 161700, 287980, 487344, 790244, 1235780, 1873200, 2763520, 3981264, 5616324, 7775940, 10586800, 14197260, 18779684, 24532904, 31684800, 40495000, 51257700, 64304604, 80007984, 98783860, 121095300, 147455840, 178433024

Offset: 1

Martin Renner, May 22 2010

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A045996, A000938.

Binomial[Range[30]^2, 3] (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,4,84,560,2300,7140,18424},30] (* Harvey P. Dale, Nov 09 2012 *)

a(n)=binomial(n^2,3) \\ Charles R Greathouse IV, Feb 19 2017

a(n) = A000938(n) + A045996(n).
a(n) = binomial(n^2,3) = 1/6*n^2*(n^2-1)*(n^2-2). - Martin Renner, May 23 2010
G.f.: 4*x^2*(1+x)*(1+13*x+x^2)/(1-x)^7. - Colin Barker, Jan 19 2012
a(1)=0, a(2)=4, a(3)=84, a(4)=560, a(5)=2300, a(6)=7140, a(7)=18424, a(n)=7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7). - Harvey P. Dale, Nov 09 2012
Sum_{n>=2} 1/a(n) = Pi^2/2 - 15/4 - 3*Pi*cot(sqrt(2)*Pi)/(2*sqrt(2)). - Amiram Eldar, Feb 17 2024

Extended by Ray Chandler, May 03 2011

Corrected by Harvey P. Dale, Nov 09 2012

A189346 Number of sets of four points on an n X n grid (or geoboard), exactly three of which are collinear.

Original entry on oeis.org

0, 0, 48, 532, 3088, 11340, 33824, 83288, 183344, 364304, 681872, 1194100, 1992976, 3182332, 4941360, 7420640, 10874720, 15539952, 21812720, 30011924, 40650368, 54187196, 71463440, 92990296, 119675712, 152314920, 192393872, 240690060

Offset: 1

Martin Renner, Apr 20 2011

nonn

The four points build a triangle on an n X n grid, with one of them located on a side of the triangle.

The number of sets of four points with the three collinear points in a horizontal or vertical line is 2*n^2*(n-1)*binomial(n,3) = 4*A090448(n). The number of sets of four points with the three collinear points in a diagonal line of slope 1 is 2*n*(n-1)*binomial(n,3) + 4*Sum_{k=3..n-1}(n^2-k)*binomial(k,3). The sum of these two values is a lower bound for this sequence. - Nathaniel Johnston, Apr 23 2011

Martin Renner, Table of n, a(n) for n = 1..76

Cf. A000938, A175383, A178256, A189345.

A189346 := proc(n)local a,b,j,k,l,m,s,slopes,num,den,tot: tot := 0: slopes := {}: for b from 1 to ceil(n/2)-1 do for a from 0 to b do slopes := slopes union {a/b}: od: od: for s from 1 to nops(slopes) do num := numer(slopes[s]): den := denom(slopes[s]): if(num = 0)then tot := tot + 2*n^2*(n-1)*binomial(n,3): elif(num = den)then tot := tot + 2*(2*add(binomial(k,3)*(n^2-k), k=3..n) - binomial(n,3)*(n^2 - n)): else for j from 1 to n - 2*den do for k from 1 to n - 2*num do tot := tot + 4*(n^2 - 3): for l from 1 to n do for m from 1 to n do if((not l = j or not m = k) and (not l = j + den or not m = k + num) and (not l = j + 2*den or not m = k + 2*num) and (m - k)*den = num*(l - j))then tot := tot - 4: fi: od: od: od: od: fi: od: return tot: end:
seq(A189346(n),n=1..15); # Nathaniel Johnston, Apr 23 2011

a(6)-a(28) from Nathaniel Johnston, Apr 23 2011

A057566 Number of collinear triples in a 3 X n rectangular grid.

Original entry on oeis.org

0, 1, 2, 8, 20, 43, 78, 130, 200, 293, 410, 556, 732, 943, 1190, 1478, 1808, 2185, 2610, 3088, 3620, 4211, 4862, 5578, 6360, 7213, 8138, 9140, 10220, 11383, 12630, 13966, 15392, 16913, 18530, 20248, 22068, 23995, 26030, 28178, 30440, 32821, 35322

Offset: 0

John W. Layman, Oct 04 2000

Paolo Xausa, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Second differences give A047264. Third differences are periodic {5, 1, 5, 1, ...} and form A010686. See A000938 for the n X n grid.

LinearRecurrence[{3, -2, -2, 3, -1}, {0, 1, 2, 8, 20}, 50] (* Paolo Xausa, Feb 22 2024 *)

Conjecture: a(n) = 5*floor((2n^3 - 3n^2 - n)/24) + floor((2(n-1)^3 - 3(n-1)^2 - (n-1))/24) + n, which fits all of the listed terms.
From R. J. Mathar, May 23 2010: (Start)
a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5) = n^3/2 - n^2 + n + (1-(-1)^n)/4.
G.f.: x*(1 - x + 4*x^2 + 2*x^3)/((1+x)*(x-1)^4). (End)

A241222 Number of collinear point triples on a centered hexagonal grid of size n.

Original entry on oeis.org

0, 3, 69, 390, 1314, 3441, 7503, 14388, 25692, 42471, 66417, 100194, 145206, 204429, 280971, 377400, 496608, 642891, 821925, 1034742, 1288602, 1587009, 1933695, 2339100, 2802804, 3334983, 3942585, 4627002, 5404542, 6278661, 7252539, 8332968, 9537456

Offset: 1

Martin Renner, Apr 17 2014

nonn

A centered hexagonal grid of size n is a grid with A003215(n-1) points forming a hexagonal lattice.

			For n = 2 the points are on the three diagonals through the center of the hexagon as following:
    . .     . *     * .
   * * *   . * .   . * .
    . .     * .     . *

Andrew Howroyd, Table of n, a(n) for n = 1..200
Eric Weisstein's World of Mathematics, Hex Number.

Cf. A000938, A241220.

c(n,s,fmin,fmax)={sum(k=1+s, n, max(0, fmax(k-s)-max(fmin(k)-1,if(k-2*s>0,fmax(k-2*s)))))}
b(n, u, v)={c(2*n-1, u, i->max(0,i-n)+1+i\u*v, i->min(i,n)+n-1+i\u*v)}
gm(n)={my(v=vector(n)); for(g=2, n, v[g]=binomial(g+1, 3) - sum(k=2, g-1, v[k]*min(k, g-k+1))); v}
a(n)={my(gmv=gm(n-1)); 3*(binomial(2*n-1,3) + 2*sum(k=0,n-2,binomial(n+k,3)) + sum(u=1, 2*n-3, sum(v=1, 2*n-2-u, my(m=gmv[gcd(u,v)]);if(m>0, m*b(n,u,v), 0))))} \\ Andrew Howroyd, Sep 18 2017

a(7) from Martin Renner, May 31 2014
a(8)-a(22) from Giovanni Resta, May 31 2014
Terms a(23) and beyond from Andrew Howroyd, Sep 18 2017

cp's OEIS Frontend

A157882 Number of collinear point-triples in the n X n X n cube.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Extensions

A045996 Number of triangles in an n X n grid (or geoplane).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A000769 No-3-in-line problem: number of inequivalent ways of placing 2n points on an n X n grid so that no 3 are in a line.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Extensions

A334704 Triangle read by rows: T(n,k) (1 <= k <= n) = number of ways to choose three collinear points from an n X k grid of points.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A235453 Triangle T(n, k) = Number of non-equivalent (mod D_4) ways to arrange k indistinguishable points on an n X n square grid so that no three of them are collinear. Triangle read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A178256 Number of ways to choose four collinear points from an n X n grid.

Views

Author

Keywords

Examples

Links

Crossrefs

A178208 Number of ways to choose three points in an (n X n)-grid (or geoplane).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A189346 Number of sets of four points on an n X n grid (or geoboard), exactly three of which are collinear.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Extensions

A057566 Number of collinear triples in a 3 X n rectangular grid.

Views