A225828 -id:A225828 - cp's OEIS frontend

A225812 Irregular triangle read by rows: T(n,k) is the number of binary pattern classes in the (4,n)-rectangular grid with k '1's and (4n-k) '0's: two patterns are in same class if one of them can be obtained by a reflection or 180-degree rotation of the other.

1, 1, 2, 4, 2, 1, 1, 2, 10, 14, 22, 14, 10, 2, 1, 1, 4, 22, 60, 139, 208, 252, 208, 139, 60, 22, 4, 1, 1, 4, 36, 140, 476, 1092, 2044, 2860, 3270, 2860, 2044, 1092, 476, 140, 36, 4, 1, 1, 6, 56, 294, 1253, 3912, 9808, 19464, 31706, 42116, 46448, 42116, 31706

Offset: 0

Yosu Yurramendi, María Merino, Jul 30 2013

The length of row n is 4*n+1.
Sum of rows (see example) gives A225828.
This triangle is to A225828 as Losanitsch's triangle A034851 is to A005418, triangle A226048 to A225826, and triangle A226290 to A225827.
Also the number of equivalence classes of ways of placing k 1 X 1 tiles in an n X 4 rectangle under all symmetry operations of the rectangle. - Christopher Hunt Gribble, Apr 24 2015

			Irregular triangle:
1
1 2  4   2   1
1 2 10  14  22   14   10    2    1
1 4 22  60 139  208  252  208  139   60   22    4   1
1 4 36 140 476 1092 2044 2860 3270 2860 2044 1092 476 140 36  4  1 ...

Yosu Yurramendi and María Merino, Rows n = 0..26 of irregular triangle, flattened

Cf. A225826, A225827, A225828, A005418, A034851, A226048.

T[n_, k_] := (Binomial[4n, k] + If[EvenQ[k], 2 Binomial[2n, k/2], 0] + Sum[Binomial[4 Mod[n, 2], k - 2i] Binomial[4 Quotient[n, 2], i], {i, 0, Quotient[k, 2]}])/4; Table[T[n, k], {n, 0, 5}, {k, 0, 4n}] // Flatten (* Jean-François Alcover, Oct 06 2017, after Andrew Howroyd *)

T(n,k)={(binomial(4*n,k) + if(k%2==0,2*binomial(2*n,k/2),0) + sum(i=0,k\2,binomial(4*(n%2),k-2*i)*binomial(4*(n\2),i)))/4}
for(n=0,4,for(k=0,4*n, print1(T(n,k), ", "));print) \\ Andrew Howroyd, May 30 2017

Definition corrected by María Merino, May 19 2017

A228022 Irregular triangle read by rows: T(n,k) is the number of binary pattern classes in the (5,n)-rectangular grid with k '1's and (5n-k) '0's: two patterns are in same class if one of them can be obtained by a reflection or 180-degree rotation of the other.

Original entry on oeis.org

1, 1, 3, 6, 6, 3, 1, 1, 3, 15, 32, 60, 66, 60, 32, 15, 3, 1, 1, 6, 34, 129, 371, 794, 1310, 1675, 1675, 1310, 794, 371, 129, 34, 6, 1, 1, 6, 56, 294, 1253, 3912, 9808, 19464, 31706, 42116, 46448, 42116, 31706, 19464, 9808, 3912, 1253, 294, 56, 6, 1

Offset: 0

Yosu Yurramendi, María Merino, Aug 03 2013

The length of row n is 5*n+1.
Sum of rows (see example) gives A225829.
This triangle is to A225828 as Losanitsch's triangle A034851 is to A005418, triangle A226048 to A225826, triangle A226290 to A225827, and triangle A225812 to A225828.
Also the number of equivalence classes of ways of placing k 1 X 1 tiles in an n X 5 rectangle under all symmetry operations of the rectangle. - Christopher Hunt Gribble, Apr 24 2015

			Irregular triangle:
1
1 3  6   6    3    1
1 3 15  32   60   66   60    32    15     3     1
1 6 34 129  371  794 1310  1675  1675  1310   794   371   129
  34 6 1
1 6 56 294 1253 3912 9808 19464 31706 42116 46448 42116 31706
  19464 9808 3912 1253 294 56 6 1
...

Yosu Yurramendi and María Merino, Rows n = 0..31 of irregular triangle, flattened

Cf. A225826-A225829, A005418, A034851, A226048, A226290, A225812.

Definition corrected by María Merino, May 22 2017

A228165 Irregular triangle read by rows: T(n,k) is the number of binary pattern classes in the (6,n)-rectangular grid with k '1's and (6n-k) '0's: two patterns are in same class if one of them can be obtained by a reflection or 180-degree rotation of the other.

Original entry on oeis.org

1, 1, 3, 9, 10, 9, 3, 1, 1, 3, 21, 55, 135, 198, 246, 198, 135, 55, 21, 3, 1, 1, 6, 48, 218, 813, 2196, 4767, 8070, 11139, 12300, 11139, 8070, 4767, 2196, 813, 218, 48, 6, 1, 1, 6, 78, 506, 2706, 10626, 33814, 86526, 184239, 326876, 490908, 624036, 676732, 624036, 490908, 326876, 184239, 86526, 33814, 10626, 2706, 506, 78, 6, 1

Offset: 0

Yosu Yurramendi, María Merino, Aug 14 2013

The length of row n is 6*n+1.
Sum of rows (see example) gives A225830.
This triangle is to A225828 as Losanitsch's triangle A034851 is to A005418, triangle A226048 to A225826, triangle A226290 to A225827, triangle A225812 to A225828, and triangle A228022 to A225829.
Also the number of equivalence classes of ways of placing k 1 X 1 tiles in an n X 6 rectangle under all symmetry operations of the rectangle. - Christopher Hunt Gribble, Apr 24 2015

			Irregular triangle:
1
1 3  9 10   9    3    1
1 3 21 55 135  198  246  198   135    55    21    3    1
1 6 48 18 813 2196 4767 8070 11139 12300 11139 8070 4767 2196 813 218 48 6 1

Yosu Yurramendi, María Merino, Rows n = 0..20 of irregular triangle, flattened

Cf. A225826-A225830, A005418, A034851, A226048, A226290, A225812, A228022.

Definition corrected by María Merino, May 22 2017

A228166 Irregular triangle read by rows: T(n,k) is the number of binary pattern classes in the (7,n)-rectangular grid with k '1's and (7n-k) '0's: two patterns are in same class if one of them can be obtained by a reflection or 180-degree rotation of the other.

Original entry on oeis.org

1, 1, 4, 12, 19, 19, 12, 4, 1, 1, 4, 28, 94, 266, 508, 777, 868, 777, 508, 266, 94, 28, 4, 1, 1, 8, 65, 363, 1574, 5231, 13826, 29454, 51408, 74130, 88900, 88900, 74130, 51408, 29454, 13826, 5231, 1574, 363, 65, 8, 1, 1, 8, 106, 832, 5199, 24648, 94524, 296296, 777997, 1727440, 3282774, 5369832, 7608483, 9362256, 10032648, 9362256

Offset: 0

Yosu Yurramendi, María Merino, Aug 14 2013

The length of row n is 7*n+1.
Sum of rows (see example) gives A225831.
This triangle is to A225828 as Losanitsch's triangle A034851 is to A005418, triangle A226048 to A225826, triangle A226290 to A225827, triangle A225812 to A225828, triangle A228022 to A225829, and triangle A228165 to A225830.
Also the number of equivalence classes of ways of placing k 1 X 1 tiles in an n X 7 rectangle under all symmetry operations of the rectangle. - Christopher Hunt Gribble, Apr 24 2015

			Irregular triangle:
1
1 4 12  19   19   12     4     1
1 4 28  94  266  508   777   868   777   508   266    94    28     4     1
1 8 65 363 1574 5231 13826 29454 51408 74130 88900 88900 74130 51408 29454 13826 5231 1574 363 65 8 1

Yosu Yurramendi, María Merino, Rows n = 0..20 of irregular triangle, flattened

Cf. A225826-A225829, A005418, A034851, A226048, A226290, A225812, A228022, A228165.

Definition corrected by María Merino, May 22 2017

A228167 Irregular triangle read by rows: T(n,k) is the number of binary pattern classes in the (8,n)-rectangular grid with k '1's and (8n-k) '0's: two patterns are in the same class if one of them can be obtained by a reflection or 180-degree rotation of the other.

Original entry on oeis.org

1, 1, 4, 16, 28, 38, 28, 16, 4, 1, 1, 4, 36, 140, 476, 1092, 2044, 2860, 3270, 2860, 2044, 1092, 476, 140, 36, 4, 1, 1, 8, 84, 536, 2770, 10808, 34116, 87144, 185071, 328208, 492392, 625968, 678524, 625968, 492392, 328208, 185071, 87144, 34116, 10808, 2770, 536, 84, 8, 1

Offset: 0

Yosu Yurramendi, María Merino, Aug 14 2013

The length of row n is 8*n+1.
Sum of rows (see example) gives A225832.
This triangle is to A225828 as Losanitsch's triangle A034851 is to A005418, triangle A226048 to A225826, triangle A226290 to A225827, triangle A225812 to A225828, triangle A228022 to A225829, triangle A228165 to A225830, and triangle A228166 to A225831.
Also the number of equivalence classes of ways of placing k 1 X 1 tiles in an n X 8 rectangle under all symmetry operations of the rectangle. - Christopher Hunt Gribble, Feb 28 2014

			Irregular triangle:
1
1 4  16   28   38    28     16      4       1
1 4  36  140  476  1092   2044   2860    3270    2860     2044   1092 ...
1 8  84  536 2770 10808  34116  87144  185071  328208   492392 625968   678524   625968    492392   ...

Yosu Yurramendi and María Merino, Rows n = 0..20 of irregular triangle, flattened

Cf. A225826-A225831, A005418, A034851, A226048, A226290, A225812, A228022, A228165, A228166.

Definition corrected by María Merino, May 22 2017

A228168 Irregular triangle read by rows: T(n,k) is the number of binary pattern classes in the (9,n)-rectangular grid with k '1's and (9n-k) '0's: two patterns are in same class if one of them can be obtained by a reflection or 180-degree rotation of the other.

Original entry on oeis.org

1, 1, 5, 20, 44, 66, 66, 44, 20, 5, 1, 1, 5, 45, 208, 792, 2156, 4704, 7984, 11034, 12190, 11034, 7984, 4704, 2156, 792, 208, 45, 5, 1, 1, 10, 106, 785, 4554, 20556, 74828, 223498, 557599, 1175574, 2114550, 3266415, 4354188, 5023464, 5023464, 4354188

Offset: 0

Yosu Yurramendi, María Merino, Aug 14 2013

The length of row n is 9*n+1.
Sum of rows (see example) gives A225832.
This triangle is to A225828 as Losanitsch's triangle A034851 is to A005418, triangle A226048 to A225826, triangle A226290 to A225827, triangle A225812 to A225828, triangle A228022 to A225829, triangle A228165 to A225830, triangle A228166 to A225831, and triangle A228167 to A225832.
Also the number of equivalence classes of ways of placing k 1 X 1 tiles in an n X 9 rectangle under all symmetry operations of the rectangle. - Christopher Hunt Gribble, Apr 27 2015

			Irregular triangle:
1
1  5  20  44   66    66    44     20      5       1
1  5  45 208  792  2156  4704   7984  11034   12190   11034    7984  ...
1 10 106 785 4554 20556 74828 223498 557599 1175574 2114550 3266415 4354188 5023464 5023464 4354188 3266415 2114550 1175574 ...

Yosu Yurramendi and María Merino, Rows n = 0..18 of irregular triangle, flattened
Christopher Hunt Gribble, C++ program

Cf. A225826-A225832, A005418, A034851, A226048, A226290, A225812, A228022, A228165, A228166, A228167.

Definition corrected by María Merino, May 22 2017

A228169 Irregular triangle read by rows: T(n,k) is the number of binary pattern classes in the (10,n)-rectangular grid with k '1's and (10n-k) '0's: two patterns are in the same class if one of them can be obtained by a reflection or 180-degree rotation of the other.

Original entry on oeis.org

1, 1, 5, 25, 60, 110, 126, 110, 60, 25, 5, 1, 1, 5, 55, 285, 1245, 3876, 9780, 19380, 31650, 41990, 46378, 41990, 31650, 19380, 9780, 3876, 1245, 285, 55, 5, 1, 1, 10, 130, 1070, 7080, 36102, 149785, 511260, 1468215, 3584050, 7523956, 13672690, 21646530, 29964990, 36386895, 38808456, 36386895, 29964990

Offset: 0

Yosu Yurramendi, María Merino, Aug 14 2013

The length of row n is 10*n+1.
Sum of rows (see example) gives A225834.
This triangle is to A225828 as Losanitsch's triangle A034851 is to A005418, triangle A226048 to A225826, triangle A226290 to A225827, triangle A225812 to A225828, triangle A228022 to A225829, and triangle A228165 to A225830, triangle A228166 to A225831, triangle A228167 to A225832, and triangle A228168 to A225833.
Also the number of equivalence classes of ways of placing k 1 X 1 tiles in an n X 10 rectangle under all symmetry operations of the rectangle. - Christopher Hunt Gribble, Mar 01 2014

			Irregular triangle:
1
1  5  25   60  110   126    110     60      25       5       1
1  5  55  285 1245  3876   9780  19380   31650   41990   46378   41990...
1 10 130 1070 7080 36102 149785 511260 1468215 3584050 7523956 13672690 21646530 29964990 36386895 38808456 36386895 29964990 21646530 ...

Yosu Yurramendi, María Merino, Rows n = 0..16 of irregular triangle, flattened

Cf. A225826-A225833, A005418, A034851, A226048, A226290, A225812, A228022, A228165-A228168.

Definition corrected by María Merino, May 22 2017

cp's OEIS Frontend

A225812 Irregular triangle read by rows: T(n,k) is the number of binary pattern classes in the (4,n)-rectangular grid with k '1's and (4n-k) '0's: two patterns are in same class if one of them can be obtained by a reflection or 180-degree rotation of the other.

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A228022 Irregular triangle read by rows: T(n,k) is the number of binary pattern classes in the (5,n)-rectangular grid with k '1's and (5n-k) '0's: two patterns are in same class if one of them can be obtained by a reflection or 180-degree rotation of the other.

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A228165 Irregular triangle read by rows: T(n,k) is the number of binary pattern classes in the (6,n)-rectangular grid with k '1's and (6n-k) '0's: two patterns are in same class if one of them can be obtained by a reflection or 180-degree rotation of the other.

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A228166 Irregular triangle read by rows: T(n,k) is the number of binary pattern classes in the (7,n)-rectangular grid with k '1's and (7n-k) '0's: two patterns are in same class if one of them can be obtained by a reflection or 180-degree rotation of the other.

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A228167 Irregular triangle read by rows: T(n,k) is the number of binary pattern classes in the (8,n)-rectangular grid with k '1's and (8n-k) '0's: two patterns are in the same class if one of them can be obtained by a reflection or 180-degree rotation of the other.

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Extensions

A228168 Irregular triangle read by rows: T(n,k) is the number of binary pattern classes in the (9,n)-rectangular grid with k '1's and (9n-k) '0's: two patterns are in same class if one of them can be obtained by a reflection or 180-degree rotation of the other.

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Extensions

A228169 Irregular triangle read by rows: T(n,k) is the number of binary pattern classes in the (10,n)-rectangular grid with k '1's and (10n-k) '0's: two patterns are in the same class if one of them can be obtained by a reflection or 180-degree rotation of the other.

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