A054252 -id:A054252 - cp's OEIS frontend

A236915 Number T(n,k) of equivalence classes of ways of placing k 8 X 8 tiles in an n X n square under all symmetry operations of the square; irregular triangle T(n,k), n>=8, 0<=k<=floor(n/8)^2, read by rows.

1, 1, 1, 1, 1, 3, 1, 3, 1, 6, 1, 6, 1, 10, 1, 10, 1, 15, 25, 5, 1, 1, 15, 79, 65, 14, 1, 21, 187, 377, 174, 1, 21, 351, 1365, 1234, 1, 28, 606, 3900, 6124, 1, 28, 948, 9282, 23259, 1, 36, 1426, 19726, 73204, 1, 36, 2026, 38046, 199436

Offset: 8

Christopher Hunt Gribble, Feb 01 2014

The first 16 rows of T(n,k) are:
.\ k  0      1      2      3      4
n
   1      1
   1      1
  1      3
  1      3
  1      6
  1      6
  1     10
  1     10
  1     15     25      5      1
  1     15     79     65     14
  1     21    187    377    174
  1     21    351   1365   1234
  1     28    606   3900   6124
  1     28    948   9282  23259
  1     36   1426  19726  73204
  1     36   2026  38046 199436

			T(16,3) = 5 because the number of equivalence classes of ways of placing 3 8 X 8 square tiles in an 16 X 16 square under all symmetry operations of the square is 5. The portrayal of an example from each equivalence class is:
._____________________        _____________________
|          |          |      |          |__________|
|          |          |      |          |          |
|          |          |      |          |          |
|     .    |     .    |      |     .    |          |
|          |          |      |          |     .    |
|          |          |      |          |          |
|          |          |      |          |          |
|__________|__________|      |__________|          |
|          |          |      |          |__________|
|          |          |      |          |          |
|          |          |      |          |          |
|    .     |          |      |     .    |          |
|          |          |      |          |          |
|          |          |      |          |          |
|          |          |      |          |          |
|__________|__________|      |__________|__________|
.
._____________________        _____________________
|          |          |      |          |          |
|          |__________|      |          |          |
|          |          |      |          |__________|
|     .    |          |      |     .    |          |
|          |          |      |          |          |
|          |     .    |      |          |          |
|          |          |      |          |     .    |
|__________|          |      |__________|          |
|          |          |      |          |          |
|          |__________|      |          |          |
|          |          |      |          |__________|
|     .    |          |      |     .    |          |
|          |          |      |          |          |
|          |          |      |          |          |
|          |          |      |          |          |
|__________|__________|      |__________|__________|
.
._____________________
|          |          |
|          |          |
|          |          |
|     .    |__________|
|          |          |
|          |          |
|          |          |
|__________|     .    |
|          |          |
|          |          |
|          |          |
|     .    |__________|
|          |          |
|          |          |
|          |          |
|__________|__________|

Christopher Hunt Gribble, Rows n = 8..23, flattened
Christopher Hunt Gribble, C++ program

Cf. A054252, A236679, A236560, A236757, A236800, A236829, A236865, A236936, A236939.

It appears that:
T(n,0) = 1, n>= 8
T(n,1) = (floor((n-8)/2)+1)*(floor((n-8)/2+2))/2, n >= 8
T(c+2*8,2) = A131474(c+1)*(8-1) + A000217(c+1)*floor(8^2/4) + A014409(c+2), 0 <= c < 8, c even
T(c+2*8,2) = A131474(c+1)*(8-1) + A000217(c+1)*floor((8-1)(8-3)/4) + A014409(c+2), 0 <= c < 8, c odd
T(c+2*8,3) = (c+1)(c+2)/2(2*A002623(c-1)*floor((8-c-1)/2) + A131941(c+1)*floor((8-c)/2)) + S(c+1,3c+2,3), 0 <= c < 8 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 Number T(n,k) of equivalence classes of ways of placing k 9 X 9 tiles in an n X n square under all symmetry operations of the square; irregular triangle T(n,k), n>=9, 0<=k<=floor(n/9)^2, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 3, 1, 6, 1, 6, 1, 10, 1, 10, 1, 15, 1, 15, 30, 5, 1, 1, 21, 96, 74, 14, 1, 21, 221, 413, 174, 1, 28, 417, 1525, 1234, 1, 28, 705, 4290, 6124, 1, 36, 1107, 10269, 23259, 1, 36, 1638, 21630, 73204, 1, 45, 2334, 41790, 199436

Offset: 9

Christopher Hunt Gribble, Feb 01 2014

			The first 17 rows of T(n,k) are:
.\ k  0      1      2      3      4
n
9     1      1
10    1      1
11    1      3
12    1      3
13    1      6
14    1      6
15    1     10
16    1     10
17    1     15
18    1     15     30      5      1
19    1     21     96     74     14
20    1     21    221    413    174
21    1     28    417   1525   1234
22    1     28    705   4290   6124
23    1     36   1107  10269  23259
24    1     36   1638  21630  73204
25    1     45   2334  41790 199436
.
T(18,3) = 5 because the number of equivalence classes of ways of placing 3 9 X 9 square tiles in an 18 X 18 square under all symmetry operations of the square is 5.

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

Cf. A054252, A236679, A236560, A236757, A236800, A236829, A236865, A236915, A236939.

It appears that:
T(n,0) = 1, n>= 9
T(n,1) = (floor((n-9)/2)+1)*(floor((n-9)/2+2))/2, n >= 9
T(c+2*9,2) =  A131474(c+1)*(9-1) + A000217(c+1)*floor(9^2/4) + A014409(c+2), 0 <= c < 9, c even
T(c+2*9,2) = A131474(c+1)*(9-1) + A000217(c+1)*floor((9-1)(9-3)/4) + A014409(c+2), 0 <= c < 9, c odd
T(c+2*9,3) = (c+1)(c+2)/2(2*A002623(c-1)*floor((9-c-1)/2) + A131941(c+1)*floor((9-c)/2)) + S(c+1,3c+2,3), 0 <= c < 9 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 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.

Original entry on oeis.org

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

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

A343875 Array read by antidiagonals: T(n,k) is the number of n X n nonnegative integer matrices with sum of elements equal to k, up to rotations and reflections.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 3, 3, 1, 0, 1, 4, 11, 3, 1, 0, 1, 8, 31, 24, 6, 1, 0, 1, 10, 84, 113, 55, 6, 1, 0, 1, 16, 198, 528, 410, 99, 10, 1, 0, 1, 20, 440, 2003, 2710, 1091, 181, 10, 1, 0, 1, 29, 904, 6968, 15233, 10488, 2722, 288, 15, 1, 0, 1, 35, 1766, 21593, 75258, 82704, 34399, 5806, 461, 15, 1

Offset: 0

Andrew Howroyd, May 06 2021

			Array begins:
=====================================================
n\k | 0  1   2    3     4      5       6        7
----+------------------------------------------------
  0 | 1  0   0    0     0      0       0        0 ...
  1 | 1  1   1    1     1      1       1        1 ...
  2 | 1  1   3    4     8     10      16       20 ...
  3 | 1  3  11   31    84    198     440      904 ...
  4 | 1  3  24  113   528   2003    6968    21593 ...
  5 | 1  6  55  410  2710  15233   75258   331063 ...
  6 | 1  6  99 1091 10488  82704  563864  3376134 ...
  7 | 1 10 181 2722 34399 360676 3235551 25387944 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325

Rows n=0..3 are A000007, A000012, A005232, A054343.
Columns 0..1 are A000012, A008805(n-1).
Cf. A054252 (binary case), A318795, A343097, A343874.

U(n,s) = {(s(1)^(n^2) + s(1)^(n%2)*(2*s(4)^(n^2\4) + s(2)^(n^2\2)) + 2*s(1)^n*s(2)^(n*(n-1)/2) + 2*(s(1)^(n%2)*s(2)^(n\2))^n )/8}
T(n,k)={polcoef(U(n,i->1/(1-x^i) + O(x*x^k)), k)}

A231655 Triangle T(n, k) read by rows giving number of non-equivalent ways to choose k points in an equilateral triangle grid of side n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 4, 6, 4, 2, 1, 1, 3, 10, 25, 41, 48, 41, 25, 10, 3, 1, 1, 4, 22, 87, 244, 526, 870, 1110, 1110, 870, 526, 244, 87, 22, 4, 1, 1, 5, 41, 238, 1029, 3450, 9147, 19524, 34104, 49231, 59038, 59038, 49231, 34104, 19524, 9147, 3450, 1029, 238

Offset: 0

Heinrich Ludwig, Nov 14 2013

Number of orbits under dihedral group D_6 of order 6. - N. J. A. Sloane, Sep 12 2019

			Triangle T(n, k) is irregularly shaped: 0 <= k <= n*(n+1)/2+1. The first row corresponds to n = 1, the first column corresponds to k = 0. Rows are palindromic.
  1,  1;
  1,  1,  1,  1;
  1,  2,  4,  6,  4,  2,  1;
  1,  3, 10, 25, 41, 48, 41, 25, 10,  3,  1;
  ...
There are T(3, 2) = 4 nonisomorphic choices of 2 points (X) in an equilateral triangle grid of side 3:
      X       .       .       X
     . .     X X     . .     X .
    . X .   . . .   X . X   . . .

Heinrich Ludwig, Table of n, a(n) for n = 0..173

Columns k=1..5 are A001399, A227327, A230723, A231653, A231654.

Cf. A054252, A283113, A289709, A326611.

A242279 Number of inequivalent (mod D_4) ways four checkers can be placed on an n X n board.

Original entry on oeis.org

1, 23, 252, 1666, 7509, 26865, 79920, 209096, 491425, 1064575, 2150076, 4104738, 7458437, 13005041, 21857984, 35598880, 56353185, 87019191, 131364700, 194364050, 282314901, 403316353, 567402672, 787201416, 1078078209, 1459020095, 1952782300, 2587048786, 3394568325

Offset: 2

Heinrich Ludwig, May 10 2014

Heinrich Ludwig, Table of n, a(n) for n = 2..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. A054252, A008805, A014409, A082966, A242358, A019318, A054772.

CoefficientList[Series[x^2*(1 + 19*x + 161*x^2 + 697*x^3 + 1446*x^4 + 2070*x^5 + 1422*x^6 + 766*x^7 + 105*x^8 + 31*x^9 + x^10 + x^11) / ((1-x)^9 * (1+x)^5), {x, 0, 20}], x] (* Vaclav Kotesovec, May 10 2014 *)
LinearRecurrence[{4,-1,-16,19,20,-45,0,45,-20,-19,16,1,-4,1},{0,0,1,23,252,1666,7509,26865,79920,209096,491425,1064575,2150076,4104738},40] (* Harvey P. Dale, May 06 2018 *)

a(n) = (n^8 - 6*n^6 + 40*n^4 - 48*n^3 + 16*n^2 + IF(MOD(n, 2) = 1)*(14*n^4 - 48*n^3 + 34*n^2 - 3))/192.
G.f.: x^2*(1 + 19*x + 161*x^2 + 697*x^3 + 1446*x^4 + 2070*x^5 + 1422*x^6 + 766*x^7 + 105*x^8 + 31*x^9 + x^10 + x^11) / ((1-x)^9 * (1+x)^5). - Vaclav Kotesovec, May 10 2014
a(n) = A054772(n, 4), n >= 2. - Wolfdieter Lang, Oct 03 2016

A242358 Number of inequivalent (mod D_4) ways five checkers can be placed on an n X n board.

Original entry on oeis.org

23, 567, 6814, 47358, 239511, 954226, 3207212, 9414828, 24862239, 60136329, 135311658, 286229762, 574460495, 1101240084, 2028333848, 3605765688, 6211552455, 10402472811, 16984387958, 27099325638, 42342870823, 64905898662, 97761436356, 144885584740, 211543443215

Offset: 3

Heinrich Ludwig, May 11 2014

Heinrich Ludwig, Table of n, a(n) for n = 3..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. A054252, A008805, A014409, A082966, A242279, A054772.

Drop[CoefficientList[Series[x^3*(-23 - 452*x - 4071*x^2 - 16016*x^3 - 40397*x^4 - 59335*x^5 - 61954*x^6 - 38236*x^7 - 17221*x^8 - 3614*x^9 - 623*x^10 + 20*x^11 + x^12 + x^13)/((x-1)^11*(x+1)^6), {x, 0, 20}], x],3] (* Vaclav Kotesovec, May 11 2014 *)

a(n) = (n^10 - 10*n^8 + 35*n^6 + 52*n^5 - 210*n^4 + 140*n^3 - 56*n^2 + 48*n + IF(MOD(n, 2) = 1)*(52*n^5 - 145*n^4 + 140*n^3 - 80*n^2 + 48*n - 15))/960.
G.f.: x^3*(-23 - 452*x - 4071*x^2 - 16016*x^3 - 40397*x^4 - 59335*x^5 - 61954*x^6 - 38236*x^7 - 17221*x^8 - 3614*x^9 - 623*x^10 + 20*x^11 + x^12 + x^13)/((x-1)^11*(x+1)^6). - Vaclav Kotesovec, May 11 2014
a(n) = A054772(n, 5), n >=3. - Wolfdieter Lang, Oct 03 2016

A331462 Triangle read by rows: T(n,k) is the number of n X n binary matrices with k=0..n^2 ones forming a polyomino, under action of dihedral group of the square D_4.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 1, 0, 3, 2, 5, 6, 10, 8, 7, 3, 1, 0, 3, 4, 8, 17, 33, 68, 119, 195, 261, 300, 257, 169, 66, 20, 3, 1, 0, 6, 6, 16, 32, 82, 189, 470, 1076, 2422, 5010, 9732, 17145, 27399, 38680, 47560, 49325, 41872, 27864, 14095, 5280, 1470, 302, 48, 6, 1

Offset: 0

Jean-Luc Manguin, Jan 17 2020

By forming a polyomino it is meant that there is at least one 1 and that all the 1's are connected horizontally or vertically.

			Triangle begins:
  0;
  0, 1;
  0, 1, 1, 1, 1;
  0, 3, 2, 5, 6, 10, 8, 7, 3, 1;
  0, 3, 4, 8, 17, 33, 68, 119, 195, 261, 300, 257, 169, 66, 20, 3, 1;
  ...

Jean-Luc Manguin, Table of n, a(n) for n = 0..97
Jean-Luc Manguin, Illustration of a(3) and comparison with A268416

Cf. A054252, A268416.

cp's OEIS Frontend

A236915 Number T(n,k) of equivalence classes of ways of placing k 8 X 8 tiles in an n X n square under all symmetry operations of the square; irregular triangle T(n,k), n>=8, 0<=k<=floor(n/8)^2, read by rows.

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A236936 Number T(n,k) of equivalence classes of ways of placing k 9 X 9 tiles in an n X n square under all symmetry operations of the square; irregular triangle T(n,k), n>=9, 0<=k<=floor(n/9)^2, read by rows.

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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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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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A343875 Array read by antidiagonals: T(n,k) is the number of n X n nonnegative integer matrices with sum of elements equal to k, up to rotations and reflections.

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A231655 Triangle T(n, k) read by rows giving number of non-equivalent ways to choose k points in an equilateral triangle grid of side n.

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A242279 Number of inequivalent (mod D_4) ways four checkers can be placed on an n X n board.

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A242358 Number of inequivalent (mod D_4) ways five checkers can be placed on an n X n board.

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