A014409 -id:A014409 - cp's OEIS frontend

A054252 Triangle T(n,k) of n X n binary matrices with k=0..n^2 ones under action of dihedral group of the square D_4.

1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 8, 16, 23, 23, 16, 8, 3, 1, 1, 3, 21, 77, 252, 567, 1051, 1465, 1674, 1465, 1051, 567, 252, 77, 21, 3, 1, 1, 6, 49, 319, 1666, 6814, 22475, 60645, 136080, 256585, 410170, 559014, 652048, 652048, 559014, 410170, 256585, 136080

Offset: 0

Vladeta Jovovic, May 04 2000

From Geoffrey Critzer, Feb 19 2013: (Start)
Cycle indices for n=2,3,4,5 respectively are:
(1/8)(s[1]^4 + 2*s[1]^2*s[2] + 3*s[2]^2 + 2*s[4]).
(1/8)(s[1]^9 + 4*s[1]^3*s[2]^3 + s[1]s[2]^4 + 2*s[1]*s[4]^2).
(1/8)(s[1]^16 + 2*s[1]^4*s[2]^6 + 2*s[4]^4 + 3*s[2]^8).
(1/8)(s[1]^25 + 4*s[1]^5*s[2]^10 + 2*s[1]*s[4]^6 + s[1]*s[2]^12).
(End)
Also the number of equivalence classes of ways of placing k 1 X 1 tiles in an n X n square under all symmetry operations of the square. - Christopher Hunt Gribble, Feb 17 2014
From Wolfdieter Lang, Oct 03 2016: (Start)
The cycle index G(n) for a square n X n grid with squares coming in two colors with k squares of one color is for the D_4 group (with 8 elements R(90)^j, S R(90)^j, j=0..3)
  (s[1]^(n^2) + s[2]^(n^2/2) +2*s[4]^(n^2/4))/8 + (s[2]^(n^2/2) + s[1]^n*s[2]^((n^2-n)/2))/4 if n is even,
  s[1]*((s[1]^(n^2-1) + s[2]^((n^2-1)/2) + 2*s[4]^((n^2-1)/4))/8) + s[1]^n*s[2]^(n*(n-1)/2)/2 if n is odd.
See the above comment by Geoffrey Critzer for n=2..5.
The figure counting series is c(x) = 1 + x for coloring, say black and white.
Therefore the counting series is C(n,x) = G(n) with substitution s[2^j] = c(x^(2*j)) = 1 + x^(2^j) for j=0,1,2. Row n gives the coefficients of C(n,x) in rising (or falling) order.  This follows from Pólya's counting theorem.  See the Harary-Palmer reference, p. 42, eq. (2.4.6), and eq. (2.2.11) with n=4 on p. 37 for the cycle index of D_4.
(End)

			T(3,2) = 8 because there are 8 nonisomorphic 3 X 3 binary matrices with two ones under action of D_4:
  [0 0 0] [0 0 0] [0 0 0] [0 0 0]
  [0 0 0] [0 0 0] [0 0 1] [0 0 1]
  [0 1 1] [1 0 1] [0 1 0] [1 0 0]
---------------------------------
  [0 0 0] [0 0 0] [0 0 0] [0 0 1]
  [0 1 0] [0 1 0] [1 0 1] [0 0 0]
  [0 0 1] [0 1 0] [0 0 0] [1 0 0]
Triangle T(n,k) begins:
1;
1, 1;
1, 1, 2,  1,  1;
1, 3, 8, 16, 23, 23, 16, 8, 3, 1;

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 42, (2.4.6), p. 37, (2.2.11).

Heinrich Ludwig, Rows n = 0..16, flattened
Index entries for sequences related to groups

Cf. A014409, A019318, A054247 (row sums), A054772.

(* As a triangle *) Prepend[Prepend[Table[CoefficientList[CycleIndexPolynomial[
GraphData[{"Grid", {n, n}}, "AutomorphismGroup"],Table[Subscript[s, i], {i, 1, 4}]] /. Table[Subscript[s, i] -> 1 + x^i, {i, 1, 4}], x], {n, 2, 10}], {1, 1}], {1}] // Grid (* Geoffrey Critzer, Aug 09 2016 *)

def T(n, k):
    if n == 0 or k == 0 or k == n*n:
        return 1
    grid = graphs.Grid2dGraph(n, n)
    m = grid.automorphism_group().cycle_index().expand(2, 'b, w')
    b, w = m.variables()
    return m.coefficient({b: k, w: n*n-k})
[T(n, k) for n in range(6) for k in range(n*n + 1)] # Freddy Barrera, Nov 23 2018

A236679 Number T(n,k) of equivalence classes of ways of placing k 2 X 2 tiles in an n X n square under all symmetry operations of the square; irregular triangle T(n,k), n>=2, 0<=k<=floor(n/2)^2, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 4, 2, 1, 1, 3, 13, 20, 14, 1, 6, 37, 138, 277, 273, 143, 39, 7, 1, 1, 6, 75, 505, 2154, 5335, 7855, 6472, 2756, 459, 1, 10, 147, 1547, 10855, 50021, 153311, 311552, 416825, 361426, 200996, 71654, 16419, 2363, 211, 11, 1, 1, 10, 246, 3759, 39926, 291171

Offset: 2

Christopher Hunt Gribble, Jan 29 2014

Changing the offset from 2 to 1, this is also the sequence: "Triangle read by rows: T(n,k) is the number of nonequivalent ways to place k non-attacking kings on an n X n board." (For if each king is represented by a 2 X 2 tile with the king in the upper left corner, the kings do not attack each other.) For example, with offset 1, T(4,3) = 20 because there are 20 nonequivalent ways to place 3 kings on a 4 X 4 chessboard so that no king threatens any other. - Heinrich Ludwig and N. J. A. Sloane, Dec 21 2016

It appears that rows 2n and 2n-1 both contain n^2 + 1 entries. Rotations and reflections of placements are not counted. If they are to be counted, see A193580. - Heinrich Ludwig, Dec 11 2016

			T(4,2) = 4 because the number of equivalence classes of ways of placing 2 2 X 2 square tiles in a 4 X 4 square under all symmetry operations of the square is 4. The portrayal of an example from each equivalence class is:
._______        _______        _______        _______
| . | . |      | . |___|      | . |   |      |_______|
|___|___|      |___| . |      |___|___|      | . | . |
|       |      |   |___|      |   | . |      |___|___|
|_______|      |_______|      |___|___|      |_______|
The first 6 rows of T(n,k) are:
.\ k  0    1    2    3    4    5    6    7    8    9
n
2     1    1
3     1    1
4     1    3    4    2    1
5     1    3   13   20   14
6     1    6   37  138  277  273  143   39    7    1
7     1    6   75  505 2154 5335 7855 6472 2756  459

Heinrich Ludwig, Table of n, a(n) for n = 2..107
Christopher Hunt Gribble, C++ program

Cf. A054252, A236560, A236757, A236800, A236829, A236865, A236915, A236936, A236939.
Row sums give A275869.
Columns 2..7: A279111, A279112, A279113, A279114, A279115, A279116.
Diagonal T(n,n) is A279117.
Cf. A193580.

It appears that:
T(n,0) = 1, n>= 2
T(n,1) = (floor((n-2)/2)+1)*(floor((n-2)/2+2))/2, n >= 2
T(c+2*2,2) = A131474(c+1)*(2-1) + A000217(c+1)*floor(2^2/4) + A014409(c+2), 0 <= c < 2, c even
T(c+2*2,2) = A131474(c+1)*(2-1) + A000217(c+1)*floor((2-1)(2-3)/4) + A014409(c+2), 0 <= c < 2, c odd
T(c+2*2,3) = (c+1)(c+2)/2(2*A002623(c-1)*floor((2-c-1)/2) + A131941(c+1)*floor((2-c)/2)) + S(c+1,3c+2,3), 0 <= c < 2 where
S(c+1,3c+2,3) =
A054252(2,3),  c = 0
A236679(5,3),  c = 1

More terms from Heinrich Ludwig, Dec 11 2016 (The former entry A279118 from Heinrich Ludwig was merged into this entry by N. J. A. Sloane, Dec 21 2016)

A082963 Number of n X n 0-1 matrices with half 1's and half 0's (rounded up/down if odd).

Original entry on oeis.org

1, 1, 2, 23, 1674, 652048, 1134460910, 7900674292378, 229078019084673798, 26549036304190836144544, 12611418068196090318131968752, 23955745839516317585042064530077352, 185026624806098273753009169783707528668060

Offset: 0

Vladeta Jovovic, May 27 2003

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..50
James Grime, Maths Problem: Complete Noughts and Crosses (Burnside's Lemma)

Cf. A054247, A054252, A014409, A019318.

C(n,f)={(f(1)^(n^2) + 2*f(1)^((n%2)*n)*f(2)^((n\2)*n) + 2*f(1)^n*f(2)^binomial(n,2) + f(1)^(n%2)*f(2)^(n^2\2) + 2*f(1)^(n%2)*f(4)^(floor(n/2)*ceil(n/2)))/8}
a(n)={polcoef(C(n, k->1 + x^k), n^2\2)} \\ Andrew Howroyd, Feb 01 2020

a(n) = A054252(n, floor(n^2/2)).

Terms a(12) and beyond from Andrew Howroyd, Feb 01 2020

A236560 Number T(n,k) of equivalence classes of ways of placing k 3 X 3 tiles in an n X n square under all symmetry operations of the square; irregular triangle T(n,k), n>=3, 0<=k<=floor(n/3)^2, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 3, 6, 2, 1, 1, 6, 21, 29, 14, 1, 6, 53, 161, 174, 1, 10, 111, 665, 1713, 1549, 608, 107, 11, 1, 1, 10, 201, 1961, 9973, 24267, 29437, 17438, 4756, 459

Offset: 3

Christopher Hunt Gribble, Jan 30 2014

The first 8 rows of T(n,k) are:
.\ k  0     1     2     3     4     5     6     7     8     9
n
   1     1
   1     1
   1     3
   1     3     6     2     1
   1     6    21    29    14
   1     6    53   161   174
   1    10   111   665  1713  1549   608   107    11     1
  1    10   201  1961  9973 24267 29437 17438  4756   459

			T(6,2) = 6 because the number of equivalence classes of ways of placing 2 3 X 3 square tiles in a 6 X 6 square under all symmetry operations of the square is 6. The portrayal of an example from each equivalence class is:
.___________      ___________      ___________
|     |     |    |     |_____|    |     |     |
|  .  |  .  |    |  .  |     |    |  .  |_____|
|_____|_____|    |_____|  .  |    |_____|     |
|           |    |     |_____|    |     |  .  |
|           |    |           |    |     |_____|
|___________|    |___________|    |_____|_____|
.
.___________      ___________      ___________
|     |     |    |_____ _____|    |_____      |
|  .  |     |    |     |     |    |     |_____|
|_____|_____|    |  .  |  .  |    |  .  |     |
|     |     |    |_____|_____|    |_____|  .  |
|     |  .  |    |           |    |     |_____|
|_____|_____|    |___________|    |_____|_____|

Christopher Hunt Gribble, C++ program

Cf. A054252, A236679, A236757, A236800, A236829, A236865, A236915, A236936, A236939.

It appears that:
T(n,0) = 1, n>= 3
T(n,1) = (floor((n-3)/2)+1)*(floor((n-3)/2+2))/2, n >= 3
T(c+2*3,2) = A131474(c+1)*(3-1) + A000217(c+1)*floor(3^2/4) + A014409(c+2), 0 <= c < 3, c even
T(c+2*3,2) = A131474(c+1)*(3-1) + A000217(c+1)*floor((3-1)(3-3)/4) + A014409(c+2), 0 <= c < 3, c odd
T(c+2*3,3) = (c+1)(c+2)/2(2*A002623(c-1)*floor((3-c-1)/2) + A131941(c+1)*floor((3-c)/2)) + S(c+1,3c+2,3), 0 <= c < 3 where
S(c+1,3c+2,3) =
A054252(2,3),  c = 0
A236679(5,3),  c = 1
A236560(8,3),  c = 2

A236757 Number T(n,k) of equivalence classes of ways of placing k 4 X 4 tiles in an n X n square under all symmetry operations of the square; irregular triangle T(n,k), n>=4, 0<=k<=floor(n/4)^2, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 3, 1, 6, 9, 3, 1, 1, 6, 29, 35, 14, 1, 10, 75, 209, 174, 1, 10, 147, 765, 1234, 1, 15, 270, 2340, 7639, 6169, 1893, 242, 17, 1, 1, 15, 438, 5806, 34342, 79821, 80722, 36569, 7106, 459

Offset: 4

Christopher Hunt Gribble, Jan 30 2014

The first 10 rows of T(n,k) are:
.\ k  0     1     2     3     4     5     6     7     8     9
n
   1     1
   1     1
   1     3
   1     3
   1     6     9     3     1
   1     6    29    35    14
  1    10    75   209   174
  1    10   147   765  1234
  1    15   270  2340  7639  6169  1893   242    17     1
  1    15   438  5806 34342 79821 80722 36569  7106   459

			T(8,3) = 3 because the number of equivalence classes of ways of placing 3 4 X 4 square tiles in an 8 X 8 square under all symmetry operations of the square is 3. The portrayal of an example from each equivalence class is:
._____________          _____________          _____________
|      |      |        |      |______|        |      |      |
|   .  |   .  |        |   .  |      |        |   .  |______|
|      |      |        |      |   .  |        |      |      |
|______|______|        |______|      |        |______|   .  |
|      |      |        |      |______|        |      |      |
|   .  |      |        |   .  |      |        |   .  |______|
|      |      |        |      |      |        |      |      |
|______|______|        |______|______|        |______|______|

Christopher Hunt Gribble, C++ program

Cf. A054252, A236679, A236560, A236800, A236829, A236865, A236915, A236936, A236939.

It appears that:
T(n,0) = 1, n>= 4
T(n,1) = (floor((n-4)/2)+1)*(floor((n-4)/2+2))/2, n >= 4
T(c+2*4,2) = A131474(c+1)*(4-1) + A000217(c+1)*floor(4^2/4) + A014409(c+2), 0 <= c < 4, c even
T(c+2*4,2) = A131474(c+1)*(4-1) + A000217(c+1)*floor((4-1)(4-3)/4) + A014409(c+2), 0 <= c < 4, c odd
T(c+2*4,3) = (c+1)(c+2)/2(2*A002623(c-1)*floor((4-c-1)/2) + A131941(c+1)*floor((4-c)/2)) + S(c+1,3c+2,3), 0 <= c < 4 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 Number T(n,k) of equivalence classes of ways of placing k 5 X 5 tiles in an n X n square under all symmetry operations of the square; irregular triangle T(n,k), n>=5, 0<=k<=floor(n/5)^2, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 3, 1, 6, 1, 6, 12, 3, 1, 1, 10, 40, 44, 14, 1, 10, 97, 245, 174, 1, 15, 193, 925, 1234, 1, 15, 339, 2640, 6124, 1, 21, 555, 6617, 27074, 19336, 4785, 461, 23, 1

Offset: 5

Christopher Hunt Gribble, Jan 31 2014

The first 11 rows of T(n,k) are:
.\ k  0     1     2     3     4     5     6     7     8     9
n
   1     1
   1     1
   1     3
   1     3
   1     6
  1     6    12     3     1
  1    10    40    44    14
  1    10    97   245   174
  1    15   193   925  1234
  1    15   339  2640  6124
  1    21   555  6617 27074 19336  4785   461    23     1

			T(10,3) = 3 because the number of equivalence classes of ways of placing 3 5 X 5 square tiles in an 10 X 10 square under all symmetry operations of the square is 3. The portrayal of an example from each equivalence class is:
._______________      _______________      _______________
|       |       |    |       |_______|    |       |       |
|       |       |    |       |       |    |       |_______|
|   .   |   .   |    |   .   |       |    |   .   |       |
|       |       |    |       |   .   |    |       |       |
|_______|_______|    |_______|       |    |_______|   .   |
|       |       |    |       |_______|    |       |       |
|       |       |    |       |       |    |       |_______|
|   .   |       |    |   .   |       |    |   .   |       |
|       |       |    |       |       |    |       |       |
|_______|_______|    |_______|_______|    |_______|_______|

Christopher Hunt Gribble, C++ program

Cf. A054252, A236679, A236560, A236757, A236829, A236865, A236915, A236936, A236939.

It appears that:
T(n,0) = 1, n>= 5
T(n,1) = (floor((n-5)/2)+1)*(floor((n-5)/2+2))/2, n >= 5
T(c+2*5,2) = A131474(c+1)*(5-1) + A000217(c+1)*floor(5^2/4) + A014409(c+2), 0 <= c < 5, c even
T(c+2*5,2) = A131474(c+1)*(5-1) + A000217(c+1)*floor((5-1)(5-3)/4) + A014409(c+2), 0 <= c < 5, c odd
T(c+2*5,3) = (c+1)(c+2)/2(2*A002623(c-1)*floor((5-c-1)/2) + A131941(c+1)*floor((5-c)/2)) + S(c+1,3c+2,3), 0 <= c < 5 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 Number T(n,k) of equivalence classes of ways of placing k 6 X 6 tiles in an n X n square under all symmetry operations of the square; irregular triangle T(n,k), n>=6, 0<=k<=floor(n/6)^2, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 3, 1, 6, 1, 6, 1, 10, 16, 4, 1, 1, 10, 51, 50, 14, 1, 15, 125, 293, 174, 1, 15, 239, 1065, 1234, 1, 21, 423, 3075, 6124, 1, 21, 672, 7371, 23259, 1, 28, 1030, 16093, 81480, 51615, 10596, 808, 31, 1

Offset: 6

Christopher Hunt Gribble, Jan 31 2014

The first 13 rows of T(n,k) are:
.\ k  0     1     2     3     4     5     6     7     8     9
n
   1     1
   1     1
   1     3
   1     3
  1     6
  1     6
  1    10    16     4     1
  1    10    51    50    14
  1    15   125   293   174
  1    15   239  1065  1234
  1    21   423  3075  6124
  1    21   672  7371 23259
  1    28  1030 16093 81480 51615 10596   808    31     1

			T(12,3) = 4 because the number of equivalence classes of ways of placing 3 6 X 6 square tiles in a 12 X 12 square under all symmetry operations of the square is 4. The portrayal of an example from each equivalence class is:
._________________          _________________
|        |        |        |        |________|
|        |        |        |        |        |
|    .   |    .   |        |    .   |        |
|        |        |        |        |    .   |
|        |        |        |        |        |
|________|________|        |________|        |
|        |        |        |        |________|
|        |        |        |        |        |
|    .   |        |        |    .   |        |
|        |        |        |        |        |
|        |        |        |        |        |
|________|________|        |________|________|
.
._________________          _________________
|        |        |        |        |        |
|        |________|        |        |        |
|    .   |        |        |    .   |________|
|        |        |        |        |        |
|        |    .   |        |        |        |
|________|        |        |________|    .   |
|        |        |        |        |        |
|        |________|        |        |        |
|    .   |        |        |    .   |________|
|        |        |        |        |        |
|        |        |        |        |        |
|________|________|        |________|________|

Christopher Hunt Gribble, C++ program

Cf. A054252, A236679, A236560, A236757, A236800, A236865, A236915, A236936, A236939.

It appears that:
T(n,0) = 1, n>= 6
T(n,1) = (floor((n-6)/2)+1)*(floor((n-6)/2+2))/2, n >= 6
T(c+2*6,2) = A131474(c+1)*(6-1) + A000217(c+1)*floor(6^2/4) + A014409(c+2), 0 <= c < 6, c even
T(c+2*6,2) = A131474(c+1)*(6-1) + A000217(c+1)*floor((6-1)(6-3)/4) + A014409(c+2), 0 <= c < 6, c odd
T(c+2*6,3) = (c+1)(c+2)/2(2*A002623(c-1)*floor((6-c-1)/2) + A131941(c+1)*floor((6-c)/2)) + S(c+1,3c+2,3), 0 <= c < 6 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 Number T(n,k) of equivalence classes of ways of placing k 7 X 7 tiles in an n X n square under all symmetry operations of the square; irregular triangle T(n,k), n>=7, 0<=k<=floor(n/7)^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, 20, 4, 1, 1, 15, 65, 59, 14, 1, 15, 153, 329, 174, 1, 21, 295, 1225, 1234, 1, 21, 507, 3465, 6124, 1, 28, 810, 8358, 23259, 1, 28, 1214, 17710, 73204

Offset: 7

Christopher Hunt Gribble, Jan 31 2014

The first 13 rows of T(n,k) are:
.\ k  0     1     2     3     4     5     6     7     8     9
n
   1     1
   1     1
   1     3
  1     3
  1     6
  1     6
  1    10
  1    10    20     4     1
  1    15    65    59    14
  1    15   153   329   174
  1    21   295  1225  1234
  1    21   507  3465  6124
  1    28   810  8358 23259
  1    28  1214 17710 73204

			T(14,3) = 4 because the number of equivalent classes of ways of placing 3 7 X 7 square tiles in an 14 X 14 square under all symmetry operations of the square is 4. The portrayal of an example from each equivalence class is:
.___________________          ___________________
|         |         |        |         |_________|
|         |         |        |         |         |
|         |         |        |         |         |
|    .    |    .    |        |    .    |         |
|         |         |        |         |    .    |
|         |         |        |         |         |
|_________|_________|        |_________|         |
|         |         |        |         |_________|
|         |         |        |         |         |
|         |         |        |         |         |
|    .    |         |        |    .    |         |
|         |         |        |         |         |
|         |         |        |         |         |
|_________|_________|        |_________|_________|
.
.___________________          ___________________
|         |         |        |         |         |
|         |_________|        |         |         |
|         |         |        |         |_________|
|    .    |         |        |    .    |         |
|         |         |        |         |         |
|         |    .    |        |         |         |
|_________|         |        |_________|    .    |
|         |         |        |         |         |
|         |_________|        |         |         |
|         |         |        |         |_________|
|    .    |         |        |    .    |         |
|         |         |        |         |         |
|         |         |        |         |         |
|_________|_________|        |_________|_________|

Christopher Hunt Gribble, C++ program

Cf. A054252, A236679, A236560, A236757, A236800, A236829, A236915, A236936, A236939.

It appears that:
T(n,0) = 1, n>= 7
T(n,1) = (floor((n-7)/2)+1)*(floor((n-7)/2+2))/2, n >= 7
T(c+2*7,2) = A131474(c+1)*(7-1) + A000217(c+1)*floor(7^2/4) + A014409(c+2), 0 <= c < 7, c even
T(c+2*7,2) = A131474(c+1)*(7-1) + A000217(c+1)*floor((7-1)(7-3)/4) + A014409(c+2), 0 <= c < 7, c odd
T(c+2*7,3) = (c+1)(c+2)/2(2*A002623(c-1)*floor((7-c-1)/2) + A131941(c+1)*floor((7-c)/2)) + S(c+1,3c+2,3), 0 <= c < 7 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 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.

Original entry on oeis.org

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

cp's OEIS Frontend

A054252 Triangle T(n,k) of n X n binary matrices with k=0..n^2 ones under action of dihedral group of the square D_4.

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Sage

A236679 Number T(n,k) of equivalence classes of ways of placing k 2 X 2 tiles in an n X n square under all symmetry operations of the square; irregular triangle T(n,k), n>=2, 0<=k<=floor(n/2)^2, read by rows.

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A082963 Number of n X n 0-1 matrices with half 1's and half 0's (rounded up/down if odd).

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PARI

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A236560 Number T(n,k) of equivalence classes of ways of placing k 3 X 3 tiles in an n X n square under all symmetry operations of the square; irregular triangle T(n,k), n>=3, 0<=k<=floor(n/3)^2, read by rows.

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A236757 Number T(n,k) of equivalence classes of ways of placing k 4 X 4 tiles in an n X n square under all symmetry operations of the square; irregular triangle T(n,k), n>=4, 0<=k<=floor(n/4)^2, read by rows.

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A236800 Number T(n,k) of equivalence classes of ways of placing k 5 X 5 tiles in an n X n square under all symmetry operations of the square; irregular triangle T(n,k), n>=5, 0<=k<=floor(n/5)^2, read by rows.

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A236829 Number T(n,k) of equivalence classes of ways of placing k 6 X 6 tiles in an n X n square under all symmetry operations of the square; irregular triangle T(n,k), n>=6, 0<=k<=floor(n/6)^2, read by rows.

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A236865 Number T(n,k) of equivalence classes of ways of placing k 7 X 7 tiles in an n X n square under all symmetry operations of the square; irregular triangle T(n,k), n>=7, 0<=k<=floor(n/7)^2, read by rows.

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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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