A014409 -id:A014409 - cp's OEIS frontend

A236939 Number T(n,k) of equivalence classes of ways of placing k 10 X 10 tiles in an n X n square under all symmetry operations of the square; irregular triangle T(n,k), n>=10, 0<=k<=floor(n/10)^2, read by rows.

1, 1, 1, 1, 1, 3, 1, 3, 1, 6, 1, 6, 1, 10, 1, 10, 1, 15, 1, 15, 1, 21, 36, 6, 1, 1, 21, 113, 80, 14, 1, 28, 261, 461, 174, 1, 28, 483, 1665, 1234, 1, 36, 819, 4725, 6124, 1, 36, 1266, 11193, 23259, 1, 45, 1878, 23646, 73204

Offset: 10

Christopher Hunt Gribble, Feb 01 2014

			The first 17 rows of T(n,k) are:
.\ k  0     1     2     3     4
n
10    1     1
11    1     1
12    1     3
13    1     3
14    1     6
15    1     6
16    1    10
17    1    10
18    1    15
19    1    15
20    1    21    36     6     1
21    1    21   113    80    14
22    1    28   261   461   174
23    1    28   483  1665  1234
24    1    36   819  4725  6124
25    1    36  1266 11193 23259
26    1    45  1878 23646 73204
.
T(20,3) = 6 because the number of equivalence classes of ways of placing 3 10 X 10 square tiles in a 20 X 20 square under all symmetry operations of the square is 6.

Christopher Hunt Gribble, C++ program
Christopher Hunt Gribble, Example graphics

Cf. A054252, A236679, A236560, A236757, A236800, A236829, A236865, A236915, A236936

It appears that:
T(n,0) = 1, n>= 10
T(n,1) = (floor((n-10)/2)+1)*(floor((n-10)/2+2))/2, n >= 10
T(c+2*10,2) =  A131474(c+1)*(10-1) + A000217(c+1)*floor(10^2/4) + A014409(c+2), 0 <= c < 10, c even
T(c+2*10,2) = A131474(c+1)*(10-1) + A000217(c+1)*floor((10-1)(10-3)/4) + A014409(c+2), 0 <= c < 10, c odd
T(c+2*10,3) = (c+1)(c+2)/2(2*A002623(c-1)*floor((10-c-1)/2) + A131941(c+1)*floor((10-c)/2)) + S(c+1,3c+2,3), 0 <= c < 10 where
S(c+1,3c+2,3) =
A054252(2,3),  c = 0
A236679(5,3),  c = 1
A236560(8,3),  c = 2
A236757(11,3), c = 3
A236800(14,3), c = 4
A236829(17,3), c = 5
A236865(20,3), c = 6
A236915(23,3), c = 7
A236936(26,3), c = 8
A236939(29,3), c = 9

A279453 Triangle read by rows: T(n, k) is the number of nonequivalent ways to place k points on an n X n square grid so that no more than 2 points are on a vertical or horizontal straight line.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 3, 8, 14, 17, 9, 2, 1, 3, 21, 73, 202, 306, 285, 115, 20, 1, 6, 49, 301, 1397, 4361, 9110, 11810, 8679, 2929, 288, 1, 6, 93, 890, 6582, 34059, 126396, 326190, 568134, 624875, 390426, 111798, 8791, 1, 10, 171, 2321, 24185, 185181, 1055025

Offset: 1

Heinrich Ludwig, Dec 17 2016

Length of n-th row is A272651(n) + 1, where A272651(n) is the maximal number of points that can be placed under the condition mentioned.
Rotations and reflections of placements are not counted. If they are to be counted, see A279445.
For condition "no more than 2 points on a straight line at any angle", see A235453.

			The table begins with T(1, 0):
1 1
1 1  2   1    1
1 3  8  14   17    9    2
1 3 21  73  202  306  285   115   20
1 6 49 301 1397 4361 9110 11810 8679 2929 288
...
T(4, 3) = 73 because there are 73 nonequivalent ways to place 3 points on a 4 X 4 square grid so that no more than 2 points are on a vertical or horizontal straight line.

Heinrich Ludwig, Table of n, a(n) for n = 1..109

Row sums give A279454.
Columns 2..8: A008805, A014409, A279454, A279455, A279456, A279457, A279458.
Diagonal T(n, n) is A279452.
Cf. A279445, A235453.

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

A019318 Number of inequivalent ways of choosing n squares from an n X n board, considering rotations and reflections to be the same.

Original entry on oeis.org

1, 2, 16, 252, 6814, 244344, 10746377, 553319048, 32611596056, 2163792255680, 159593799888052, 12952412056879996, 1147044793316531040, 110066314584030859544, 11375695977099383509351, 1259843950257390597789296, 148842380543159458506703546, 18685311541775061906510072648, 2483858381692984848273972297368, 348545122958862200122401771463328

Offset: 1

Mario Velucchi (mathchess(AT)velucchi.it)

Number of n X n binary matrices with n ones under action of dihedral group of the square D_4.

			For n=3 the 16 solutions are
111 110 110 110 110 110 110 101 101 101 100 100 100 010 010 010
000 100 010 001 000 000 000 010 000 000 011 010 001 110 101 010
000 000 000 000 100 010 001 000 100 010 000 001 010 000 000 010

Mathieu Gouttenoire, Table of n, a(n) for n = 1..300
Mario Velucchi, Different Dispositions in the ChessBoard.
Mario Velucchi, Different Dispositions in the ChessBoard.

Cf. A054252 and A014409.

p[a_, b_, n_] := If[EvenQ[n], (a+b)^(n^2) + 2*(a+b)^n*(a^2 + b^2)^((n^2 - n)/2) + 3*(a^2 + b^2)^(n^2/2) + 2*(a^4 + b^4)^(n^2/4), (a+b)^(n^2) + 2*(a+b)*(a^4 + b^4)^((n^2-1)/4) + (a+b)*(a^2 + b^2)^((n^2-1)/2) + 4*(a+b)^n*(a^2 + b^2)^((n^2-n)/2)]; Table[Coefficient[p[a, 1, k], a, k]/8, {k, 1, 20}] (* Jean-François Alcover, Nov 12 2013, translated from Pari *)

{p(a,b,N) = if(N%2==0, (a+b)^(N^2) + 2*(a+b)^N*(a^2+b^2)^((N^2-N)/2) + 3*(a^2+b^2)^(N^2/2) + 2*(a^4+b^4)^(N^2/4), (a+b)^(N^2) + 2*(a+b)*(a^4+b^4)^((N^2-1)/4) + (a+b)*(a^2+b^2)^((N^2-1)/2) + 4*(a+b)^N*(a^2+b^2)^((N^2-N)/2))} for(k=1,20,print1(polcoeff(p(a,1,k),k)/8,","))

See Velucchi link or the PARI program. Note that the polynomial whose coefficient of a^k is divided by 8 differs based upon whether the term's index is even or odd.

Let A(n) = C(n^2, n); B(n) = C((n^2-(n mod 2))/2, n/2); C(n) = C((n^2-(n mod 2))/4, n/4); D(n) = Sum(p = 0 to [n/2], C((n^2-n)/2, p)*C(n, n-2p)). Then a(n) = (A(n) + 3B(n) + 2C(n) + 2D(n))/8 if n == 0 (mod 4), (A(n) + B(n) + 2C(n) + 4D(n))/8 if n == 1 (mod 4), (A(n) + 3B(n) + 2D(n))/8 if n == 2 (mod 4), (A(n) + B(n) + 4D(n))/8 if n == 3 (mod 4). - David W. Wilson, May 29 2003

More terms from Rick L. Shepherd and David W. Wilson, May 28 2003

A082966 Number of inequivalent ways (mod D_4) three checkers can be placed on an n X n board.

Original entry on oeis.org

0, 1, 16, 77, 319, 920, 2397, 5278, 10874, 20355, 36390, 61171, 99441, 154882, 235179, 346060, 499172, 702933, 974124, 1324585, 1777555, 2349116, 3070441, 3962762, 5066814, 6409975, 8044322, 10004463, 12355749, 15141190, 18441495, 22309336, 26843016, 32106217

Offset: 1

Vladeta Jovovic, May 27 2003

Erich Friedman, Illustration of initial terms
Index entries for linear recurrences with constant coefficients, signature (3,1,-11,6,14,-14,-6,11,-1,-3,1).

Cf. A014409, A054252, A242279, A242358, A054247.

Rest@ CoefficientList[Series[x^2*(x^8 - x^7 - 4 x^6 - 55 x^5 - 56 x^4 - 83 x^3 - 28 x^2 - 13 x - 1)/((x - 1)^7*(x + 1)^4), {x, 0, 34}], x] (* Michael De Vlieger, Oct 03 2016 *)

a(n) = (1/48)*(n-1)*(n^5+n^4-2*n^3+14*n^2-5*n+3) if n is odd;
a(n) = (1/48)*n*(n-1)*(n^2-n+2)*(n^2+2*n-2) if n is even.
G.f.: x^2*(x^8-x^7-4*x^6-55*x^5-56*x^4-83*x^3-28*x^2-13*x-1) / ((x-1)^7*(x+1)^4). - Colin Barker, Jul 11 2013
a(n) = A054247(n, 3) = A054247(n, n^2-3), n >= 1. - Wolfdieter Lang, Oct 03 2016
E.g.f.: (x*(3 + 24*x + 88*x^2 + 62*x^3 + 15*x^4 + x^5)*cosh(x) + (-3 + 39*x^2 + 80*x^3 + 62*x^4 + 15*x^5 + x^6)*sinh(x))/48. - Stefano Spezia, Apr 14 2022

More terms from Colin Barker, Jul 11 2013

A279447 Number of nonequivalent ways to place 3 points on an n X n square grid so that no more than 2 points are on a vertical or horizontal straight line.

Original entry on oeis.org

0, 1, 14, 73, 301, 890, 2321, 5166, 10654, 20055, 35880, 60511, 98419, 153608, 233331, 343820, 496076, 699261, 969234, 1318885, 1770185, 2340646, 3059749, 3950618, 5051786, 6393075, 8023756, 9981531, 12328239, 15110740, 18405415, 22269656, 26796504, 32055353, 38158166

Offset: 1

Heinrich Ludwig, Dec 17 2016

Column 4 of A279453.
Rotations and reflections of placements are not counted. For numbers if they are to be counted see A279437.
For condition "no more than 2 points on straight lines at any angle", see A235454.

Heinrich Ludwig, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,1,-11,6,14,-14,-6,11,-1,-3,1).

Cf. A235454, A279437, A279452, A279453, A279454.

Same problem but 2, 4..7 points: A014409, A279448, A279449, A279450, A279451.

I:=[0,1,14,73,301,890,2321,5166,10654,20055,35880]; [n le 11 select I[n] else 3*Self(n-1)+Self(n-2)-11*Self(n-3)+ 6*Self(n-4)+14*Self(n-5)-14*Self(n-6)-6*Self(n-7)+11*Self(n-8)-Self(n-9)-3*Self(n-10)+Self(n-11): n in [1..40]]; // Vincenzo Librandi, Dec 17 2016

LinearRecurrence[{3, 1, -11, 6, 14, -14, -6, 11, -1, -3, 1},{0, 1, 14, 73, 301, 890, 2321, 5166, 10654, 20055, 35880}, 35] (* Vincenzo Librandi Dec 17 2016 *)

concat(0, Vec(x^2*(1 + 11*x + 30*x^2 + 79*x^3 + 62*x^4 + 55*x^5 + 4*x^6 - x^7 - x^8) / ((1 - x)^7*(1 + x)^4) + O(x^30))) \\ Colin Barker, Dec 17 2016

a(n) = (n^6 - 5*n^4 + 14*n^3 - 14*n^2 + 4*n)/48 + IF(MOD(n, 2) = 1, 2*n^3 - 3*n^2 + 1)/16.
a(n) = 3*a(n-1) + a(n-2) - 11*a(n-3) + 6*a(n-4) + 14*a(n-5) - 14*a(n-6) - 6*a(n-7) + 11*a(n-8) - a(n-9) - 3*a(n-10) + a(n-11).
G.f.: x^2*(1 + 11*x + 30*x^2 + 79*x^3 + 62*x^4 + 55*x^5 + 4*x^6 - x^7 - x^8) / ((1 - x)^7*(1 + x)^4). - Colin Barker, Dec 17 2016

A279448 Number of nonequivalent ways to place 4 points on an n X n square grid so that no more than 2 points are on a vertical or horizontal straight line.

Original entry on oeis.org

0, 1, 17, 202, 1397, 6582, 24185, 73496, 195086, 463875, 1013505, 2061426, 3956947, 7222992, 12640817, 21312992, 34801420, 55215621, 85424721, 129174250, 191397185, 278361226, 398108777, 560635032, 778491962, 1066995527, 1445034305, 1935301746, 2565356031, 3367870500

Offset: 1

Heinrich Ludwig, Dec 18 2016

Column 5 of A279453.
Rotations and reflections of placements are not counted. For numbers if they are to be counted see A279438.
For condition "no more than 2 points on straight lines at any angle", see A235455.

Heinrich Ludwig, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-1,-16,19,20,-45,0,45,-20,-19,16,1,-4,1).

Cf. A235455, A279438, A279452, A279453, A279454.

Same problem but 2,3,5,6,7 points: A014409, A279447, A279449, A279450, A279451.

concat(0, Vec(x^2*(1 +13*x +135*x^2 +622*x^3 +1449*x^4 +2143*x^5 +1557*x^6 +781*x^7 +34*x^8 -8*x^9 -8*x^10 +x^11) / ((1 -x)^9*(1 +x)^5) + O(x^40))) \\ Colin Barker, Dec 18 2016

a(n) = (n^8 - 14*n^6 + 30*n^5 + 12*n^4 - 60*n^3 + 40*n^2)/192 + IF(MOD(n, 2) = 1, 4*n^4 - 20*n^3 + 22*n^2 - 2*n - 7)/64.
a(n) = 4*a(n-1) - a(n-2) - 16*a(n-3) + 19*a(n-4) + 20*a(n-5) - 45*a(n-6) + 45*a(n-8) - 20*a(n-9) - 19*a(n-10) + 16*a(n-11) + a(n-12) - 4*a(n-13) + a(n-14).
G.f.: x^2*(1 +13*x +135*x^2 +622*x^3 +1449*x^4 +2143*x^5 +1557*x^6 +781*x^7 +34*x^8 -8*x^9 -8*x^10 +x^11) / ((1 -x)^9*(1 +x)^5). - Colin Barker, Dec 18 2016

A279449 Number of nonequivalent ways to place 5 points on an n X n square grid so that no more than 2 points are on a vertical or horizontal straight line.

Original entry on oeis.org

0, 0, 9, 306, 4361, 34059, 185181, 777280, 2710074, 8181558, 22067973, 54285858, 123791067, 264749849, 536146569, 1035584592, 1919530804, 3430908108, 5937810417, 9984193986, 16358592141, 26181281511, 41019234245, 63028246512, 95136210222, 141264963970, 206611069197

Offset: 1

Heinrich Ludwig, Dec 18 2016

Column 6 of A279453.
Rotations and reflections of placements are not counted. For numbers if they are to be counted see A279439.
For condition "no more than 2 points on straight lines at any angle", see A235456.

Heinrich Ludwig, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-4,-20,40,16,-100,44,110,-110,-44,100,-16,-40,20,4,-5,1).

Cf. A235456, A279439, A279452, A279453, A279454.

Same problem but 2,3,4,6,7 points: A014409, A279447, A279448, A279450, A279451.

concat(vector(2), Vec(x^3*(9 +261*x +2867*x^2 +13658*x^3 +38090*x^4 +62447*x^5 +67142*x^6 +41996*x^7 +15541*x^8 +955*x^9 -761*x^10 -278*x^11 -8*x^12 +x^13) / ((1 -x)^11*(1 +x)^6) + O(x^40))) \\ Colin Barker, Dec 18 2016

a(n) = (n^10 - 30*n^8 + 90*n^7 - 27*n^6 - 218*n^5 + 340*n^4 - 340*n^3 + 376*n^2 - 192*n)/960 + IF(MOD(n, 2) = 1, 2*n^5 - 9*n^4 + 14*n^3 - 6*n^2 - 4*n + 3)/64.
a(n) = 5*a(n-1) - 4*a(n-2) - 20*a(n-3) + 40*a(n-4) + 16*a(n-5) - 100*a(n-6) + 44*a(n-7) + 110*a(n-8) - 110*a(n-9) - 44*a(n-10) + 100*a(n-11) - 16*a(n-12) - 40*a(n-13) + 20*a(n-14) + 4*a(n-15) - 5*a(n-16) + a(n-17).
G.f.: x^3*(9 +261*x +2867*x^2 +13658*x^3 +38090*x^4 +62447*x^5 +67142*x^6 +41996*x^7 +15541*x^8 +955*x^9 -761*x^10 -278*x^11 -8*x^12 +x^13) / ((1 -x)^11*(1 +x)^6). - Colin Barker, Dec 18 2016

A279450 Number of nonequivalent ways to place 6 points on an n X n square grid so that no more than 2 points are on a vertical or horizontal straight line.

Original entry on oeis.org

0, 0, 2, 285, 9110, 126396, 1055025, 6266614, 29198740, 113262680, 380775248, 1140764611, 3108667306, 7824370092, 18407341855, 40855872764, 86201399496, 173952773328, 337453762782, 631982899545, 1146743732126, 2022212701212, 3474824082125, 5831439251154, 9576836632860

Offset: 1

Heinrich Ludwig, Dec 18 2016

Column 7 of A279453.
Rotations and reflections of placements are not counted. For numbers if they are to be counted see A279440.
For condition "no more than 2 points on straight lines at any angle", see A235457.

Heinrich Ludwig, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-8,-22,69,-8,-176,168,182,-364,0,364,-182,-168,176,8,-69,22,8,-6,1).

Cf. A235457, A279440, A279452, A279453, A279454.

Same problem but 2,3,4,5,7 points: A014409, A279447, A279448, A279449, A279451.

concat(vector(2), Vec(x^3*(2 +273*x +7416*x^2 +74060*x^3 +375661*x^4 +1128403*x^5 +2194010*x^6 +2815082*x^7 +2424155*x^8 +1294751*x^9 +376028*x^10 -5296*x^11 -32173*x^12 -8195*x^13 +178*x^14 +122*x^15 +3*x^16) / ((1 -x)^13*(1 +x)^7) + O(x^40))) \\ Colin Barker, Dec 18 2016

a(n) = (n^12 - 55*n^10 + 210*n^9 + 93*n^8 - 2220*n^7 + 6052*n^6 - 8040*n^5 + 4236*n^4 + 3240*n^3 - 5872*n^2 + 2400*n)/5760 + IF(MOD(n, 2) = 1, 2*n^6 - 18*n^5 + 53*n^4 - 64*n^3 + 33*n^2 - 12*n + 5)/128.
a(n) = 6*a(n-1) - 8*a(n-2) - 22*a(n-3) + 69*a(n-4) - 8*a(n-5) - 176*a(n-6) + 168*a(n-7) + 182*a(n-8) - 364*a(n-9) + 364*a(n-11) - 182*a(n-12) - 168*a(n-13) + 176*a(n-14) + 8*a(n-15) - 69*a(n-16) + 22*a(n-17) + 8*a(n-18) - 6*a(n-19) + *a(n-20).
G.f.: x^3*(2 +273*x +7416*x^2 +74060*x^3 +375661*x^4 +1128403*x^5 +2194010*x^6 +2815082*x^7 +2424155*x^8 +1294751*x^9 +376028*x^10 -5296*x^11 -32173*x^12 -8195*x^13 +178*x^14 +122*x^15 +3*x^16) / ((1 -x)^13*(1 +x)^7). - Colin Barker, Dec 18 2016

A279451 Number of nonequivalent ways to place 7 points on an n X n square grid so that no more than 2 points are on a vertical or horizontal straight line.

Original entry on oeis.org

0, 0, 0, 115, 11810, 326190, 4444935, 38675954, 246563232, 1248782460, 5296300670, 19499431941, 63958228738, 190528987506, 523151460045, 1339408935540, 3227223506896, 7372750196952, 16069268866908, 33586411339335, 67610793877650, 131569779776182, 248290280743571

Offset: 1

Heinrich Ludwig, Dec 22 2016

Column 8 of A279453.
Rotations and reflections of placements are not counted. For numbers if they are to be counted see A279441.
For condition "no more than 2 points on straight lines at any angle", see A235458.

Heinrich Ludwig, Table of n, a(n) for n = 1..1000

Cf. A235458, A279441, A279452, A279453, A279454.

Same problem but 2..6 points: A014409, A279447, A279448, A279449, A279450.

Table[(n^14 - 91 n^12 + 420 n^11 + 693 n^10 - 10500 n^9 + 33647 n^8 - 45316 n^7 + 3682 n^6 + 62300 n^5 - 51996 n^4 - 28504 n^3 + 54384 n^2 - 18720 n)/40320 + Boole[OddQ@ n] (2 n^7 - 17 n^6 + 50 n^5 - 59 n^4 + 38 n^3 - 71 n^2 + 102 n - 45)/384, {n, 23}] (* or *)
Rest@ CoefficientList[Series[x^4*(115 + 11005 x + 245015 x^2 + 2317550 x^3 + 12037814 x^4 + 39232894 x^5 + 85494738 x^6 + 129182670 x^7 + 135873108 x^8 + 97856368 x^9 + 44499480 x^10 + 9709722 x^11 - 1359254 x^12 - 1352974 x^13 - 257282 x^14 + 13866 x^15 + 7705 x^16 + 419 x^17 + x^18)/((1 - x)^15*(1 + x)^8), {x, 0, 23}], x] (* Michael De Vlieger, Dec 23 2016 *)

concat(vector(3), Vec(x^4*(115 +11005*x +245015*x^2 +2317550*x^3 +12037814*x^4 +39232894*x^5 +85494738*x^6 +129182670*x^7 +135873108*x^8 +97856368*x^9 +44499480*x^10 +9709722*x^11 -1359254*x^12 -1352974*x^13 -257282*x^14 +13866*x^15 +7705*x^16 +419*x^17 +x^18) / ((1 -x)^15*(1 +x)^8) + O(x^30))) \\ Colin Barker, Dec 23 2016

a(n) = (n^14 -91*n^12 +420*n^11 +693*n^10 -10500*n^9 +33647*n^8 -45316*n^7 +3682*n^6 +62300*n^5 -51996*n^4 -28504*n^3 +54384*n^2 -18720*n)/40320 + IF(MOD(n, 2) = 1, 2*n^7 -17*n^6 +50*n^5 -59*n^4 +38*n^3 -71*n^2 +102*n -45)/384.

G.f.: x^4*(115 +11005*x +245015*x^2 +2317550*x^3 +12037814*x^4 +39232894*x^5 +85494738*x^6 +129182670*x^7 +135873108*x^8 +97856368*x^9 +44499480*x^10 +9709722*x^11 -1359254*x^12 -1352974*x^13 -257282*x^14 +13866*x^15 +7705*x^16 +419*x^17 +x^18) / ((1 -x)^15*(1 +x)^8). - Colin Barker, Dec 23 2016

cp's OEIS Frontend

A236939 Number T(n,k) of equivalence classes of ways of placing k 10 X 10 tiles in an n X n square under all symmetry operations of the square; irregular triangle T(n,k), n>=10, 0<=k<=floor(n/10)^2, read by rows.

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A279453 Triangle read by rows: T(n, k) is the number of nonequivalent ways to place k points on an n X n square grid so that no more than 2 points are on a vertical or horizontal straight line.

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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.

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A019318 Number of inequivalent ways of choosing n squares from an n X n board, considering rotations and reflections to be the same.

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A082966 Number of inequivalent ways (mod D_4) three checkers can be placed on an n X n board.

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A279447 Number of nonequivalent ways to place 3 points on an n X n square grid so that no more than 2 points are on a vertical or horizontal straight line.

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A279448 Number of nonequivalent ways to place 4 points on an n X n square grid so that no more than 2 points are on a vertical or horizontal straight line.

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A279449 Number of nonequivalent ways to place 5 points on an n X n square grid so that no more than 2 points are on a vertical or horizontal straight line.

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