A228022 -id:A228022 - cp's OEIS frontend

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.

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

A238009 Number T(n,k) of equivalence classes of ways of placing k 2 X 2 tiles in an n X 3 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=2, 0<=k<=floor(n/2), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 2, 4, 1, 3, 8, 3, 1, 3, 12, 8, 1, 4, 18, 22, 6, 1, 4, 24, 40, 22, 1, 5, 32, 73, 66, 10, 1, 5, 40, 112, 146, 48, 1, 6, 50, 172, 292, 174, 20, 1, 6, 60, 240, 516, 448, 116, 1, 7, 72, 335, 860, 1020, 464, 36, 1, 7, 84, 440, 1340, 2016, 1360, 256

Offset: 2

Christopher Hunt Gribble, Feb 16 2014

			The first 19 rows of T(n,k) are:
   n\k 0  1   2    3    4     5     6     7     8    9  10
   2   1  1
   3   1  1
   4   1  2   2
   5   1  2   4
   6   1  3   8    3
   7   1  3  12    8
   8   1  4  18   22    6
   9   1  4  24   40   22
  10   1  5  32   73   66    10
  11   1  5  40  112  146    48
  12   1  6  50  172  292   174    20
  13   1  6  60  240  516   448   116
  14   1  7  72  335  860  1020   464    36
  15   1  7  84  440 1340  2016  1360   256
  16   1  8  98  578 2010  3716  3400  1168    72
  17   1  8 112  728 2890  6336  7432  3840   584
  18   1  9 128  917 4046 10326 14864 10600  2920  136
  19   1  9 144 1120 5502 16016 27536 25344 10600 1280
  20   1 10 162 1368 7336 24066 48188 54992 31800 7080 272

Christopher Hunt Gribble and Andrew Howroyd, Rows n=2..60 of T(n,k) flattened (Rows n=2..20 from Christopher Hunt Gribble)
Christopher Hunt Gribble, C++ program

Cf. A034851, A226048, A102541, A226290, A228570, A225812, A238189, A238190, A228572, A228022, A231145, A231473, A231568, A232440, A228165, A238550, A238551, A238552, A228166, A238555, A238556, A228167, A238557, A238558, A238559, A228168, A238581, A238582, A238583, A228169, A238586, A238592.

T(n,k)={(2^k*binomial(n-1*k,k) + ((k%2==0)+(n%2==0||k%2==0)+(k==0)) * 2^((k+1)\2)*binomial((n-1*k-(k%2)-(n%2))/2,k\2))/4}
for(n=2,20,for(k=0,floor(n/2), print1(T(n,k), ", "));print) \\ Andrew Howroyd, May 29 2017

Corrected C++ program and xrefs added by Christopher Hunt Gribble, Apr 25 2015

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

A231145 Number T(n,k) of equivalence classes of ways of placing k 2 X 2 tiles in an n X 5 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=2, 0<=k<=n-n%2, read by rows.

Original entry on oeis.org

1, 2, 2, 1, 2, 4, 1, 4, 13, 10, 4, 1, 4, 23, 35, 23, 1, 6, 40, 101, 125, 54, 10, 1, 6, 58, 206, 403, 336, 106, 1, 8, 83, 392, 1056, 1438, 956, 240, 25, 1, 8, 109, 641, 2281, 4424, 4718, 2409, 473, 1, 10, 142, 1011, 4429, 11370, 17252, 14478, 6094, 1020, 70

Offset: 2

Christopher Hunt Gribble, Feb 23 2014

			The first 8 rows of T(n,k) are:
.\ k  0     1     2     3     4     5     6     7     8
n
2     1     2     2
3     1     2     4
4     1     4    13    10     4
5     1     4    23    35    23
6     1     6    40   101   125    54    10
7     1     6    58   206   403   336   106
8     1     8    83   392  1056  1438   956   240    25
9     1     8   109   641  2281  4424  4718  2409   473

Andrew Howroyd, Table of n, a(n) for n = 2..967 (terms n=2..71 from Christopher Hunt Gribble)
Christopher Hunt Gribble, C++ program

Cf. A034851, A226048, A102541, A226290, A238009, A228570, A225812, A238189, A238190, A228572, A228022, A231473, A231568, A232440, A228165, A238550, A238551, A238552, A228166, A238555, A238556, A228167, A238557, A238558, A238559, A228168, A238581, A238582, A238583, A228169, A238586, A238592.

Terms corrected and xrefs updated by Christopher Hunt Gribble, Apr 26 2015

A231473 Number T(n,k) of equivalence classes of ways of placing k 3 X 3 tiles in an n X 5 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=3, 0<=k<=floor(n/3), read by rows.

Original entry on oeis.org

1, 2, 1, 2, 1, 4, 1, 4, 4, 1, 6, 9, 1, 6, 18, 1, 8, 28, 10, 1, 8, 42, 28, 1, 10, 57, 76, 1, 10, 76, 140, 25, 1, 12, 96, 254, 107, 1, 12, 120, 392, 321, 1, 14, 145, 600, 731, 70, 1, 14, 174, 840, 1462, 366, 1, 16, 204, 1170, 2610, 1308, 1, 16, 238, 1540, 4350, 3416, 196

Offset: 3

Christopher Hunt Gribble, Feb 23 2014

			The first 11 rows of T(n,k) are:
.\ k    0      1      2      3      4
n
3       1      2
4       1      2
5       1      4
6       1      4      4
7       1      6      9
8       1      6     18
9       1      8     28     10
10      1      8     42     28
11      1     10     57     76
12      1     10     76    140     25
13      1     12     96    254    107

Andrew Howroyd, Table of n, a(n) for n = 3..974
Christopher Hunt Gribble, C++ program

Cf. A034851, A226048, A102541, A226290, A238009, A228570, A225812, A238189, A238190, A228572, A228022, A231145, A231568, A232440, A228165, A238550-A238552, A228166, A238555, A238556, A228167, A238557-A238559, A228168, A238581-A238583, A228169, A238586, A238592.

T[n_, k_] := ((3^k + 1)*Binomial[n - 2k, k] + Boole[EvenQ[k] || OddQ[n]]*(3^(Quotient[(k + 1), 2]) + 3^Quotient[k, 2]) Binomial[(n - 2k - Mod[n, 2])/2, Quotient[k, 2]])/4; Table[T[n, k], {n, 3, 20}, {k, 0, Floor[n/3]}] // Flatten (* Jean-François Alcover, Oct 06 2017, after Andrew Howroyd *)

T(n,k)={((3^k+1)*binomial(n-2*k,k) + (k%2==0||n%2==1) * (3^((k+1)\2)+3^(k\2)) * binomial((n-2*k-(n%2))/2,k\2))/4}
for(n=3,20,for(k=0,floor(n/3), print1(T(n,k), ", "));print) \\ Andrew Howroyd, May 29 2017

Terms corrected and xrefs updated by Christopher Hunt Gribble, Apr 26 2015

Terms a(40) and beyond from Andrew Howroyd, May 29 2017

A231568 Number T(n,k) of equivalence classes of ways of placing k 4 X 4 tiles in an n X 5 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=4, 0<=k<=floor(n/4), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 2, 1, 3, 4, 1, 4, 8, 1, 4, 12, 1, 5, 18, 3, 1, 5, 24, 8, 1, 6, 32, 22, 1, 6, 40, 40, 1, 7, 50, 73, 6, 1, 7, 60, 112, 22, 1, 8, 72, 172, 66, 1, 8, 84, 240, 146, 1, 9, 98, 335, 292, 10, 1, 9, 112, 440, 516, 48, 1, 10, 128, 578, 860, 174

Offset: 4

Christopher Hunt Gribble, Feb 23 2014

			The first 14 rows of T(n,k) are:
.\  k    0      1      2      3     4
n
4        1      1
5        1      1
6        1      2
7        1      2
8        1      3      2
9        1      3      4
10       1      4      8
11       1      4     12
12       1      5     18      3
13       1      5     24      8
14       1      6     32     22
15       1      6     40     40
16       1      7     50     73     6
17       1      7     60    112    22

Andrew Howroyd, Table of n, a(n) for n = 4..989
Christopher Hunt Gribble, C++ program

Cf. A034851, A226048, A102541, A226290, A238009, A228570, A225812, A238189, A238190, A228572, A228022, A231145, A231473, A232440, A228165, A238550, A238551, A238552, A228166, A238555, A238556, A228167, A238557, A238558, A238559, A228168, A238581, A238582, A238583, A228169, A238586, A238592.

T[n_, k_] := (2^k Binomial[n - 3k, k] + (Boole[EvenQ[k]] + Boole[EvenQ[n] || EvenQ[k]] + Boole[k == 0]) 2^Quotient[k+1, 2] Binomial[(n - 3k - Mod[k, 2] - Mod[n, 2])/2, Quotient[k, 2]])/4; Table[T[n, k], {n, 4, 20}, {k, 0, Floor[n/4]}] // Flatten (* Jean-François Alcover, Oct 06 2017, after Andrew Howroyd *)

T(n,k)={(2^k*binomial(n-3*k,k) + ((k%2==0)+(n%2==0||k%2==0)+(k==0)) * 2^((k+1)\2)*binomial((n-3*k-(k%2)-(n%2))/2,k\2))/4}
for(n=2,20,for(k=0,floor(n/4), print1(T(n,k), ", "));print) \\ Andrew Howroyd, May 29 2017

Terms extended and xrefs updated by Christopher Hunt Gribble, Apr 26 2015

Terms a(32) and beyond from Andrew Howroyd, May 29 2017

A232440 Number T(n,k) of equivalence classes of ways of placing k 5 X 5 tiles in an n X 5 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=5, 0<=k<=floor(n/5), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 3, 1, 1, 4, 2, 1, 4, 4, 1, 5, 6, 1, 5, 9, 1, 6, 12, 1, 1, 6, 16, 2, 1, 7, 20, 6, 1, 7, 25, 10, 1, 8, 30, 19, 1, 8, 36, 28, 1, 1, 9, 42, 44, 3, 1, 9, 49, 60, 9, 1, 10, 56, 85, 19, 1, 10, 64, 110, 38, 1, 11, 72, 146, 66, 1

Offset: 5

Christopher Hunt Gribble, Feb 23 2014

			The first 9 rows of T(n,k) are:
.\ k    0      1      2
n
5       1      1
6       1      1
7       1      2
8       1      2
9       1      3
10      1      3      1
11      1      4      2
12      1      4      4
13      1      5      6

Andrew Howroyd, Table of n, a(n) for n = 5..989
Christopher Hunt Gribble, C++ program

Cf. A034851, A226048, A102541, A226290, A238009, A228570, A225812, A238189, A238190, A228572, A228022, A231145, A231473, A231568, A228165, A238550-A238552, A228166, A238555, A238556, A228167, A238557-A238559, A228168, A238581-A238583, A228169, A238586, A238592.

T[n_, k_] := (Binomial[n - 4k, k] + Boole[EvenQ[k] || OddQ[n]] Binomial[(n - 4k - Mod[n, 2])/2, Quotient[k, 2]])/2; Table[T[n, k], {n, 5, 20}, {k, 0, Quotient[n, 5]}] // Flatten (* Jean-François Alcover, Oct 06 2017, after Andrew Howroyd *)

T(n,k)={(binomial(n-4*k,k) + (k%2==0||n%2==1)*binomial((n-4*k-n%2)/2,k\2))/2}
for(n=5,20,for(k=0,(n\5), print1(T(n,k), ", "));print) \\ Andrew Howroyd, May 29 2017

Terms extended and xrefs updated by Christopher Hunt Gribble, Apr 26 2015

Terms a(27) and beyond from Andrew Howroyd, May 29 2017

cp's OEIS Frontend

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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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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A238009 Number T(n,k) of equivalence classes of ways of placing k 2 X 2 tiles in an n X 3 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=2, 0<=k<=floor(n/2), read by rows.

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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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A231145 Number T(n,k) of equivalence classes of ways of placing k 2 X 2 tiles in an n X 5 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=2, 0<=k<=n-n%2, read by rows.

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A231473 Number T(n,k) of equivalence classes of ways of placing k 3 X 3 tiles in an n X 5 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=3, 0<=k<=floor(n/3), read by rows.

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A231568 Number T(n,k) of equivalence classes of ways of placing k 4 X 4 tiles in an n X 5 rectangle under all symmetry operations of the rectangle; irregular triangle T(n,k), n>=4, 0<=k<=floor(n/4), read by rows.

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