A018808 -id:A018808 - cp's OEIS frontend

A295707 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals, where A(n,k) is the number of lines through at least 2 points of an n X k grid of points.

0, 1, 1, 1, 6, 1, 1, 11, 11, 1, 1, 18, 20, 18, 1, 1, 27, 35, 35, 27, 1, 1, 38, 52, 62, 52, 38, 1, 1, 51, 75, 93, 93, 75, 51, 1, 1, 66, 100, 136, 140, 136, 100, 66, 1, 1, 83, 131, 181, 207, 207, 181, 131, 83, 1, 1, 102, 164, 238, 274, 306, 274, 238, 164, 102, 1

Offset: 1

Seiichi Manyama, Nov 26 2017

			Square array begins:
   0,  1,  1,   1,   1, ...
   1,  6, 11,  18,  27, ...
   1, 11, 20,  35,  52, ...
   1, 18, 35,  62,  93, ...
   1, 27, 52,  93, 140, ...
   1, 38, 75, 136, 207, ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened
Seppo Mustonen, On lines and their intersection points in a rectangular grid of points
Seppo Mustonen, On lines and their intersection points in a rectangular grid of points [Local copy]

Columns k=2..10 give A160842, A160843, A160844, A160845, A160846, A160847, A160848, A160849, A160850.

Main diagonal gives A018808. Reading up to the diagonal gives A107348.

A[n_, k_] := (1/2)(f[n, k, 1] - f[n, k, 2]);
f[n_, k_, m_] := Sum[If[GCD[mx/m, my/m] == 1, (n - Abs[mx])(k - Abs[my]), 0], {mx, -n, n}, {my, -k, k}];
Table[A[n - k + 1, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 04 2023 *)

A(n,k) = (1/2) * (f(n,k,1) - f(n,k,2)), where f(n,k,m) = Sum ((n-|m*x|)*(k-|m*y|)); -n < m*x < n, -k < m*y < k, (x,y)=1.

A222267 The number of distinct lines defined by an n X n X n grid of points.

Original entry on oeis.org

28, 253, 1492, 5485, 17092, 41905, 95140, 191773, 364420, 638785, 1085500, 1745389, 2743084, 4136257, 6101740, 8747821, 12377764, 17066737, 23287564, 31174813, 41276548, 53767873, 69544324, 88722973, 112450132, 140859361, 175324636

Offset: 2

Clive Tooth, Feb 13 2013

Given a cubic n X n X n grid of points, a(n) is the number of distinct lines produced by constructing a line through every pair of points.
Define the grid as consisting of the set of n^3 distinct points whose x, y and z coordinates are all integers in [0..n-1]. Assign to each grid point a distinct index j = x + n*y + n^2*z. For each pair of grid points P_A and P_B (where P_A is the one with the lower index j), let L be the line that passes through both grid points, and let S be the segment of that line from P_A to P_B. Examine each of the C(n^3,2) pairs of distinct grid points P_A and P_B; a(n) is the number of those pairs for which S does not pass through any other grid points between P_A and P_B, nor does L pass through any other grid points beyond the P_A end of S. - Jon E. Schoenfield, Sep 21 2013
Conjecture: a(n) is approximately 0.3639537*n^6, with a relative error of about 10^-5 when n is near 200. - Clive Tooth, Mar 03 2016

			Each of the 28 pairs of points on a 2 X 2 X 2 grid of points defines a distinct line, so a(2) = 28.
Of the 351 pairs of points on a 3 X 3 X 3 grid, there are only 253 distinct lines, so a(3) = 253.

Clive Tooth, Table of n, a(n) for n = 2..200 (using a method of Haukkanen & Merikoski) [Terms 2 through 70 were computed by Jon E. Schoenfield]
P. Haukkanen, J. K. Merikoski, Some formulas for numbers of line segments and lines in a rectangular grid, arXiv:1108.1041 [math.CO], 2011.
Clive Tooth, A C# class declaration

Cf. A018808, A222268 (number of intersection points of these lines).

mq[{x1_, y1_}, {x2_, y2_}] := If[x1 == x2, {x1}, {y2 - y1, x2*y1 - x1*y2}/(x2 - x1)]; two[n_] := Block[{p = Tuples[Range@n, 2]},
Length@Union@Flatten[Table[mq[p[[i]], p[[j]]], {i, 2, n^2}, {j, i - 1}], 1]]; coef[a_, b_] := Block[{d = b - a}, If[d[[1]] == 0, {0}, d *= Sign@d[[1]]/GCD @@ d; {a - d*a[[1]]/d[[1]], d}]]; a[n_] := Block[{p = Tuples[Range@n, 3]}, n*two[n] - 1 + Length@Union@ Flatten[Table[coef[p[[i]], p[[j]]], {i, 2, n^3}, {j, i - 1}], 1]]; Table[v = a[n]; Print@v; v, {n, 2, 12}] (* Giovanni Resta, Feb 14 2013 *)

a(6)-a(12) from Giovanni Resta, Feb 14 2013

a(13)-a(28) from Jon E. Schoenfield, Sep 16 2013

A333282 Triangle read by rows: T(m,n) (m >= n >= 1) = number of regions formed by drawing the line segments connecting any two of the (m+1) X (n+1) lattice points in an m X n grid of squares and extending them to the boundary of the grid.

Original entry on oeis.org

4, 16, 56, 46, 192, 624, 104, 428, 1416, 3288, 214, 942, 3178, 7520, 16912, 380, 1672, 5612, 13188, 29588, 51864, 648, 2940, 9926, 23368, 52368, 92518, 164692, 1028, 4624, 15732, 37184, 83628, 148292, 263910, 422792, 1562, 7160, 24310, 57590, 130034, 230856, 410402, 658080, 1023416

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 16 2020

Triangle gives number of nodes in graph LC(m,n) in the notation of Blomberg-Shannon-Sloane (2020).

If we only joined pairs of the 2(m+n) boundary points, we would get A331452. If we did not extend the lines to the boundary of the grid, we would get A288187. (One of the links below shows the difference between the three definitions in the case m=3, n=2.)

			Triangle begins:
4,
16, 56,
46, 192, 624,
104, 428, 1416, 3288,
214, 942, 3178, 7520, 16912,
380, 1672, 5612, 13188, 29588, 51864,
648, 2940, 9926, 23368, 52368, 92518, 164692,
1028, 4624, 15732, 37184, 83628, 148292, 263910, 422792
1562, 7160, 24310, 57590, 130034, 230856, 410402, 658080, 1023416
2256, 10336, 35132, 83116, 187376, 331484, 588618, 942808, 1466056, 2101272

Lars Blomberg, Scott R. Shannon, and N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2020). Also arXiv:2009.07918.
Seppo Mustonen, Statistical accuracy of geometric constructions, 2008.
Seppo Mustonen, Statistical accuracy of geometric constructions, 2008 [Local copy]
Seppo Mustonen, On lines and their intersection points in a rectangular grid of points, 2009
Seppo Mustonen, On lines and their intersection points in a rectangular grid of points, 2009 [Local copy]
Seppo Mustonen, On lines going through a given number of points in a rectangular grid of points, 2010
Seppo Mustonen, On lines going through a given number of points in a rectangular grid of points, 2010 [Local copy]
Scott R. Shannon, Colored illustration for T(3,2)
Scott R. Shannon, Colored illustration for T(3,2) (edge number coloring)
Scott R. Shannon, Colored illustration for T(3,3)
Scott R. Shannon, Colored illustration for T(3,3) (edge number coloring)
Scott R. Shannon, Colored illustration for T(4,4)
Scott R. Shannon, Colored illustration for T(4,4) (edge number coloring)
Scott R. Shannon, Colored illustration for T(5,5)
Scott R. Shannon, Colored illustration for T(5,5) (edge number coloring)
Scott R. Shannon, Colored illustration for T(6,3)
Scott R. Shannon, Colored illustration for T(6,3) (edge number coloring)
Scott R. Shannon, Colored illustration for T(7,4)
Scott R. Shannon, Colored illustration for T(7,4) (edge number coloring)
Scott R. Shannon, Colored illustration for T(8,2)
Scott R. Shannon, Colored illustration for T(8,2) (edge number coloring)
Scott R. Shannon, Numerical properties of these structures. Corrected version Mar 24 2020.
N. J. A. Sloane, Illustration of T(3,2) = 192. [Black lines correspond to A331452(3,2), black + red lines correspond to A288187(3,2), and black + red + blue lines to T(3,2)]
N. J. A. Sloane, Illustration of T(3,3) = 624 [Black lines correspond to A288187(3,3), and black + red lines to T(3,3)]

Cf. A288187, A331452, A333283 (edges), A333284 (vertices). Column 1 is A306302. Main diagonal is A333294.

More terms and corrections from Scott R. Shannon, Mar 21 2020

More terms from Scott R. Shannon, May 27 2021

A107348 Triangle read by rows: T(m,n) = number of different lines in a rectangular m X n array of points with integer coordinates (x,y): 0 <= x <= m, 0 <= y <= n.

Original entry on oeis.org

0, 1, 6, 1, 11, 20, 1, 18, 35, 62, 1, 27, 52, 93, 140, 1, 38, 75, 136, 207, 306, 1, 51, 100, 181, 274, 405, 536, 1, 66, 131, 238, 361, 534, 709, 938, 1, 83, 164, 299, 454, 673, 894, 1183, 1492, 1, 102, 203, 370, 563, 836, 1111, 1470, 1855, 2306

Offset: 0

Dan Dima, May 23 2005

We may assume n <= m since T(m,n)=T(n,m).

			Triangle begins
0,
1, 6,
1, 11, 20,
1, 18, 35, 62,
1, 27, 52, 93, 140,
1, 38, 75, 136, 207, 306,
1, 51, 100, 181, 274, 405, 536,
1, 66, 131, 238, 361, 534, 709, 938,
1, 83, 164, 299, 454, 673, 894, 1183, 1492,
1, 102, 203, 370, 563, 836, 1111, 1470, 1855, 2306,
...

M. A. Alekseyev, M. Basova, and N. Yu. Zolotykh, On the minimal teaching sets of two-dimensional threshold functions, SIAM J. Disc. Math. 29(1), 2015, pp. 157-165.
Les Reid, Problem #7: How Many Lines Does the Lattice of Points Generate?, Problems from the 04-05 academic year, Challenge Archive, Missouri State University's Problem Corner.

Cf. A295707 (symmetric array), A018808 (diagonal). A160842 - A160850 (columns).

VR := proc(m,n,q) local a,i,j; a:=0;
for i from -m+1 to m-1 do for j from -n+1 to n-1 do
if gcd(i,j)=q then a:=a+(m-abs(i))*(n-abs(j)); fi; od: od: a; end;
LL:=(m,n)->(VR(m,n,1)-VR(m,n,2))/2;
for m from 1 to 12 do lprint([seq(LL(m,n),n=1..m)]); od: # N. J. A. Sloane, Feb 10 2020

VR[m_, n_, q_] := Sum[If[GCD[i, j] == q, (m - Abs[i])(n - Abs[j]), 0], {i, -m + 1, m - 1}, {j, -n + 1, n - 1}];
LL[m_, n_] := (1/2)(VR[m, n, 1] - VR[m, n, 2]);
Table[LL[m, n], {m, 1, 10}, {n, 1, m}] // Flatten (* Jean-François Alcover, Jun 04 2023, after N. J. A. Sloane *)

T(0, 0) = 0; T(m, 0) = 1, m >= 1.
When both m,n -> +oo, T(m,n) / 2Cmn -> 9/(2*pi^2). - Dan Dima, Mar 18 2006
T(n,m) = A295707(n,m). - R. J. Mathar, Dec 17 2017

T(3,3) corrected and sequence extended by R. J. Mathar, Dec 17 2017

A119437 Table T(n,k) = number of lines through exactly k points of an n X n grid of points.

Original entry on oeis.org

6, 12, 8, 48, 4, 10, 108, 16, 4, 12, 248, 36, 4, 4, 14, 428, 64, 20, 4, 4, 16, 764, 100, 44, 4, 4, 4, 18, 1196, 204, 36, 24, 4, 4, 4, 20, 1900, 252, 64, 52, 4, 4, 4, 4, 22, 2668, 396, 124, 40, 28, 4, 4, 4, 4, 24, 3824, 572, 200, 20, 60, 4, 4, 4, 4, 4, 26, 5244, 780, 236, 76, 44, 32

Offset: 2

Franklin T. Adams-Watters, May 19 2006

			From _Seiichi Manyama_, Nov 26 2017: (Start)
The table starts:
  n\k|   2    3   4   5   6   7   8
  ---+------------------------------
   2 |   6;
   3 |  12,   8;
   4 |  48,   4, 10;
   5 | 108   16,  4, 12;
   6 | 248,  36,  4,  4, 14;
   7 | 428,  64, 20,  4,  4, 16;
   8 | 764, 100, 44,  4,  4,  4, 18; (End)

Seiichi Manyama, Rows n = 2..141, flattened
S. Mustonen, On lines and their intersection points in a rectangular grid of points [From _Seppo Mustonen_, Apr 18 2009]
Seppo Mustonen, On lines and their intersection points in a rectangular grid of points [Local copy]

Row sums A018808; columns A018809-A018817. See A119439 for another version.

T(n,k) = 1/2 (f(n, k+1) - 2 f(n, k) + f(n, k-1)) where f(n, k) = Sum_{-n < kx < n, -n < ky < n, gcd(x, y)=1} (n - |kx|)*(n - |ky|). [Seppo Mustonen, Apr 18 2009]

An incorrect formula removed by Seppo Mustonen, Apr 25 2009

A160842 Number of lines through at least 2 points of a 2 X n grid of points.

Original entry on oeis.org

0, 1, 6, 11, 18, 27, 38, 51, 66, 83, 102, 123, 146, 171, 198, 227, 258, 291, 326, 363, 402, 443, 486, 531, 578, 627, 678, 731, 786, 843, 902, 963, 1026, 1091, 1158, 1227, 1298, 1371, 1446, 1523, 1602, 1683, 1766, 1851, 1938, 2027, 2118, 2211, 2306, 2403, 2502

Offset: 0

Seppo Mustonen, May 28 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
S. Mustonen, On lines and their intersection points in a rectangular grid of points
Seppo Mustonen, On lines and their intersection points in a rectangular grid of points [Local copy]
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A295707, A107348.

[0,1] cat [n^2 + 2: n in [2..100]]; // G. C. Greubel, Apr 30 2018

a[n_]:=If[n<2,n,n^2+2] Table[a[n],{n,0,50}]
Join[{0,1},Range[2,50]^2+2] (* Harvey P. Dale, Feb 06 2015 *)

Vec(-x*(2*x^3-4*x^2+3*x+1) / (x-1)^3 + O(x^100)) \\ Colin Barker, May 24 2015

a(n) = n^2 + 2 = A059100(n) = A010000(n) for n > 1.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 4. - Colin Barker, May 24 2015
G.f.: -x*(2*x^3 - 4*x^2 + 3*x + 1) / (x-1)^3. - Colin Barker, May 24 2015
Sum_{n>=1} 1/a(n) = Pi * coth(sqrt(2)*Pi) / 2^(3/2) - 1/4. - Vaclav Kotesovec, May 01 2018

A187397 Expansion of -2x^4 (3x^13 +2x^12 +x^11 -6x^10 -10x^9 -6x^8 +x^7 +7x^6 +5x^5 -x^4 -8x^3 -11x^2 -8x -5) / ((x -1)^4 (x +1)^2 (x^2 +1)^2 *(x^2 +x +1)^2).

Original entry on oeis.org

0, 0, 0, 0, 10, 16, 22, 36, 54, 66, 92, 122, 156, 196, 240, 288, 366, 426, 490, 590, 698, 780, 904, 1036, 1176, 1326, 1484, 1650, 1874, 2060, 2254, 2512, 2782, 3006, 3300, 3606, 3924, 4256, 4600, 4956, 5398, 5782, 6178, 6666, 7170, 7608, 8144

Offset: 0

Sean A. Irvine, Mar 23 2011

In contrast, the number of distinct lines passing through 4 or more points in an n X n grid is given by 0, 0, 0, 10, 16, 22, 44, 74, 92, 154, 232, 326, 436, 562, 704, 998, 1268,.. = A018808(n) -A018809(n) -A018810(n) = A225606(n) -A018810(n). - David W. Wilson, Aug 05 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0, 0, 2, 2, 0, -1, -4, -1, 0, 2, 2, 0, 0, -1).

Cf. A178465, A187062.

CoefficientList[ Series[ 2x^4 (5 + 8x + 11x^2 + 8x^3 + x^4 - 5x^5 - 7x^6 - x^7 + 6x^8 + 10x^9 + 6x^10 - x^11 - 2x^12 - 3x^13)/((-1 + x)^4 (1 + x)^2 (1 + x^2)^2 (1 + x + x^2)^2), {x, 0, 43}], x] (* or *) LinearRecurrence[{0, 0, 2, 2, 0, -1, -4, -1, 0, 2, 2, 0, 0, -1}, {10, 16, 22, 36, 54, 66, 92, 122, 156, 196, 240, 288, 366, 426}, 40] (* Robert G. Wilson v, Feb 17 2014 *)

Definition replaced with Colin Barker's g.f. by R. J. Mathar, Aug 06 2013

Offset changed from 1 to 0 and a(0)=0 added by Vincenzo Librandi, Feb 19 2014

A225606 Number of distinct lines passing through 3 or more points in an n X n grid.

Original entry on oeis.org

0, 0, 0, 8, 14, 32, 58, 108, 174, 296, 406, 628, 898, 1216, 1582, 2188, 2754, 3528, 4398, 5524, 6778, 8336, 9778, 11812, 14038, 16456, 19066, 22540, 25954, 29968, 34270, 39116, 44282, 50312, 56026, 63196, 70798, 78984

Offset: 0

R. J. Mathar, Aug 06 2013

nonn

P. Haukkanen, J. K. Merikoski, Some formulas for numbers of line segments and lines in a rectangular grid, arXiv:1108.1041
R. J. Mathar, Maple program implementing A225606, A018808 and sequ. in comment of A187397

f[n_, k_] := Sum[x = kx/k; y = ky/k; If[IntegerQ[x] && IntegerQ[y] && CoprimeQ[x, y], (n - Abs[kx])(n - Abs[ky]), 0], {kx, -n + 1, n - 1}, {ky, -n + 1, n - 1}];
a[n_] := (f[n, 2] - f[n, 3])/2;
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Oct 30 2018, after Seppo Mustonen in A018808 *)

a(n) = A018808(n) - A018809(n) = A018810(n) + A018811(n) + A018812(n) + A018813(n)+....

A333283 Triangle read by rows: T(m,n) (m >= n >= 1) = number of edges formed by drawing the line segments connecting any two of the (m+1) X (n+1) lattice points in an m X n grid of squares and extending them to the boundary of the grid.

Original entry on oeis.org

8, 28, 92, 80, 320, 1028, 178, 716, 2348, 5512, 372, 1604, 5332, 12676, 28552, 654, 2834, 9404, 22238, 49928, 87540, 1124, 5008, 16696, 39496, 88540, 156504, 279100, 1782, 7874, 26458, 62818, 141386, 251136, 447870

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 16 2020

If we only joined pairs of the 2(m+n) boundary points, we would get A331454. If we did not extend the lines to the boundary of the grid, we would get A333278. (One of the links below shows the difference between the three definitions in the case m=3, n=2.)

See A333282 for a large number of colored illustrations.

			Triangle begins:
8,
28, 92,
80, 320, 1028,
178, 716, 2348, 5512,
372, 1604, 5332, 12676, 28552,
654, 2834, 9404, 22238, 49928, 87540,
1124, 5008, 16696, 39496, 88540, 156504, 279100,
1782, 7874, 26458, 62818, 141386, 251136, 447870, ...
...
T(7,7) corrected Mar 19 2020

Seppo Mustonen, Statistical accuracy of geometric constructions, 2008.
Seppo Mustonen, Statistical accuracy of geometric constructions, 2008 [Local copy]
Seppo Mustonen, On lines and their intersection points in a rectangular grid of points, 2009
Seppo Mustonen, On lines and their intersection points in a rectangular grid of points, 2009 [Local copy]
Seppo Mustonen, On lines going through a given number of points in a rectangular grid of points, 2010
Seppo Mustonen, On lines going through a given number of points in a rectangular grid of points, 2010 [Local copy]
N. J. A. Sloane, Illustration of T(3,2) = 320. [Black lines correspond to A331454(3,2), black + red lines correspond to A333278(3,2), and black + red + blue lines to T(3,2)]
N. J. A. Sloane, Illustration of T(3,3) = 1028 [Black lines correspond to A288187(3,3), and black + red lines to T(3,3)]

Cf. A288187, A331452, A333278, A331454, A333282 (regions), A333284 (vertices). Column 1 is A331757.

More terms and corrections from Scott R. Shannon, Mar 21 2020

A178465 Expansion of -2x^2(-3-2x+x^2-x^3-2x^4+x^5) / ( (1+x)^2*(x-1)^4 ).

Original entry on oeis.org

0, 0, 6, 16, 36, 66, 114, 176, 264, 370, 510, 672, 876, 1106, 1386, 1696, 2064, 2466, 2934, 3440, 4020, 4642, 5346, 6096, 6936, 7826, 8814, 9856, 11004, 12210, 13530, 14912, 16416, 17986, 19686, 21456, 23364, 25346, 27474, 29680, 32040, 34482

Offset: 0

Sean A. Irvine, Mar 23 2011

nonn

Cf. A187062, A187397, A061804, A018808.

CoefficientList[ Series[ 2x^2 (3 + 2x - x^2 + x^3 + 2x^4 - x^5)/((1 + x)^2 (x - 1)^4), {x, 0, 42}], x] (* Robert G. Wilson v, Feb 17 2014 *)

def A178465(n): return n+(m:=n&1)+(n*(n**2-m)>>1) if n != 1 else 0 # Chai Wah Wu, Aug 30 2022

For n even, a(n) = n*(2+n^2)/2 = A061804(n/2). For n>1 and odd, a(n)=(n+1)*(n^2-n+2)/2 = 2*A212133((n+1)/2).

a(n) = (2-2*(-1)^n+(3+(-1)^n)*n+2*n^3)/4 for n>1. [Colin Barker, Feb 18 2013]

Discrepancy with A018808 resolved. David W. Wilson, Aug 05 2013
First line of formulas corrected. R. J. Mathar, Aug 05 2013
Prepended a(0)=0, Joerg Arndt, Feb 19 2014

cp's OEIS Frontend

A295707 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals, where A(n,k) is the number of lines through at least 2 points of an n X k grid of points.

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A222267 The number of distinct lines defined by an n X n X n grid of points.

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A333282 Triangle read by rows: T(m,n) (m >= n >= 1) = number of regions formed by drawing the line segments connecting any two of the (m+1) X (n+1) lattice points in an m X n grid of squares and extending them to the boundary of the grid.

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A107348 Triangle read by rows: T(m,n) = number of different lines in a rectangular m X n array of points with integer coordinates (x,y): 0 <= x <= m, 0 <= y <= n.

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Maple

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A119437 Table T(n,k) = number of lines through exactly k points of an n X n grid of points.

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A160842 Number of lines through at least 2 points of a 2 X n grid of points.

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Magma

Mathematica

PARI

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A187397 Expansion of -2*x^4 *(3*x^13 +2*x^12 +x^11 -6*x^10 -10*x^9 -6*x^8 +x^7 +7*x^6 +5*x^5 -x^4 -8*x^3 -11*x^2 -8*x -5) / ((x -1)^4 *(x +1)^2 *(x^2 +1)^2 *(x^2 +x +1)^2).

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Mathematica

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A225606 Number of distinct lines passing through 3 or more points in an n X n grid.

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A187397 Expansion of -2x^4 (3x^13 +2x^12 +x^11 -6x^10 -10x^9 -6x^8 +x^7 +7x^6 +5x^5 -x^4 -8x^3 -11x^2 -8x -5) / ((x -1)^4 (x +1)^2 (x^2 +1)^2 *(x^2 +x +1)^2).

A178465 Expansion of -2x^2(-3-2x+x^2-x^3-2x^4+x^5) / ( (1+x)^2*(x-1)^4 ).