A225910 -id:A225910 - cp's OEIS frontend

A225826 Number of binary pattern classes in the (2,n)-rectangular grid: two patterns are in same class if one of them can be obtained by a reflection or 180-degree rotation of the other.

1, 3, 7, 24, 76, 288, 1072, 4224, 16576, 66048, 262912, 1050624, 4197376, 16785408, 67121152, 268468224, 1073790976, 4295098368, 17180065792, 68720001024, 274878693376, 1099513724928, 4398049656832, 17592194433024, 70368756760576

Offset: 0

Yosu Yurramendi, May 16 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Gregory Emmett Coxson and Jon Carmelo Russo, Enumeration and Generation of PSL Equivalence Classes for Quad-Phase Codes of Even Length, IEEE Transactions on Aerospace and Electronic Systems, Year: 2017, Volume: 53, Issue: 4, p. 1907-1915.
Vincent Pilaud and V. Pons, Permutrees, arXiv preprint arXiv:1606.09643 [math.CO], 2016.
Index entries for linear recurrences with constant coefficients, signature (4,4,-16).

Cf. A005418 = Number of binary pattern classes in the (1,n)-rectangular grid, A225826 to A225834 are the numbers of binary pattern classes in the (m,n)-rectangular grid, 1 < m < 11, A132390 is the sequence when the 90-degree rotation for pattern equivalence is allowed. So, only a(2) is different (communicated by Jon E. Schoenfield). See A054247 for (n,n)-grids.

A225910 is the table of (m,n)-rectangular grids.

[2^(n-3)*(2^(n+1)-(-1)^n+7): n in [0..25]]; // Vincenzo Librandi, Sep 03 2013

LinearRecurrence[{4, 4, -16}, {1, 3, 7}, 30] (* Bruno Berselli, May 17 2013 *)
CoefficientList[Series[(1 - x - 9 x^2) / ((1 - 2 x) (1 + 2 x) (1 - 4 x)), {x, 0, 33}], x] (* Vincenzo Librandi, Sep 03 2013 *)

a(n) = 4*a(n-1) + 4*a(n-2) - 16*a(n-3) with n>2, a(0)=1, a(1)=3, a(2)=7 (communicated by Jon E. Schoenfield).
a(n) = 2^(n-3)*(2^(n+1) - (-1)^n + 7).
G.f.: (1-x-9*x^2)/((1-2*x)*(1+2*x)*(1-4*x)).

A225834 Number of binary pattern classes in the (10,n)-rectangular grid: 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, 528, 262912, 268713984, 274878693376, 281475261923328, 288230376957018112, 295147905471410601984, 302231454904481927397376, 309485009821644135887536128, 316912650057058194799105933312, 324518553658427033027930681769984, 332306998946228969090642893525221376

Offset: 0

Yosu Yurramendi, May 16 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (1024,1024,-1048576).

A005418 is the number of binary pattern classes in the (1,n)-rectangular grid.
A225826 to A225834  are the numbers of binary pattern classes in the (m,n)-rectangular grid, 1 < m < 11 .
A225910 is the table of (m,n)-rectangular grids.

[2^(5*n-3)*(2^(5*n+1)-(2^5-1)*(-1)^n+2^5+5): n in [0..20]]; // Vincenzo Librandi, Sep 04 2013

CoefficientList[Series[(1 - 496 x - 278784 x^2) / ((1 - 32 x) (1 + 32 x) (1 - 1024 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Sep 04 2013 *)

a(n) = 2^10*a(n-1) + 2^10*a(n-2) - (2^10)^2*a(n-3), with n>2 , a(0)=1, a(1)=528, a(2)=262912.
a(n) = 2^(5n-3)*(2^(5n+1)-(2^5-1)*(-1)^n+2^5+5).
G.f.: (1-496*x-278784*x^2)/((1-32*x)*(1+32*x)*(1-1024*x)).

A225827 Number of binary pattern classes in the (3,n)-rectangular grid: 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, 6, 24, 168, 1120, 8640, 66816, 529920, 4212736, 33632256, 268713984, 2148630528, 17184194560, 137456517120, 1099579785216, 8796367749120, 70369826308096, 562954298720256, 4503616874348544, 36028866141093888, 288230651566489600, 2305844111946547200

Offset: 0

Yosu Yurramendi, May 16 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-24,-96,256).

A005418 is the number of binary pattern classes in the (1,n)-rectangular grid.
A225826 to A225834  are the numbers of binary pattern classes in the (m,n)-rectangular grid, 1 < m < 11.
A225910 is the table of (m,n)-rectangular grids.

I:=[1,6,24,168]; [n le 4 select I[n] else 12*Self(n-1)-24*Self(n-2)-96*Self(n-3)+256*Self(n-4): n in [1..30]]; // Vincenzo Librandi, Sep 04 2013

LinearRecurrence[{12, -24, -96, 256}, {1, 6, 24, 168}, 20] (* Bruno Berselli, May 17 2013 *)
CoefficientList[Series[(1 - 6 x - 24 x^2 + 120 x^3) / ((1 - 4 x) (1 - 8 x) (1 - 8 x^2)), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 04 2013 *)

a(n) = 8*a(n-1) + 8*a(n-2) - 64*a(n-3) - 2^(2n-3) with n>2, with a(0)=1, a(1)=6, a(2)=24.
a(n) = 2^(3n/2-1)*(2^(3n/2-1) + 2^(n/2-1) + 1) if n is even,
a(n) = 2^((3*n-1)/2-1)*(2^((3*n-1)/2) + 2^((n-1)/2) + 3) if n is odd.
G.f.: (1-6*x-24*x^2+120*x^3)/((1-4*x)*(1-8*x)*(1-8*x^2)). [Bruno Berselli, May 17 2013]

More terms from Vincenzo Librandi, Sep 04 2013

A225828 Number of binary pattern classes in the (4,n)-rectangular grid: 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, 10, 76, 1120, 16576, 263680, 4197376, 67133440, 1073790976, 17180262400, 274878693376, 4398052802560, 70368756760576, 1125900007505920, 18014398710808576, 288230377762324480, 4611686021648613376, 73786976320608010240, 1180591620768950910976, 18889465931890897715200

Offset: 0

Yosu Yurramendi, May 16 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..800
Index entries for linear recurrences with constant coefficients, signature (16,16,-256).

A005418 is the number of binary pattern classes in the (1,n)-rectangular grid.
A225826 to A225834 are the numbers of binary pattern classes in the (m,n)-rectangular grid, 1 < m < 11.
A225910 is the table of (m,n)-rectangular grids.

I:=[1, 10, 76]; [n le 3 select I[n] else 16*Self(n-1)+16*Self(n-2)-256*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Sep 04 2013

Table[2^(2 n - 3) (2^(2 n + 1) - 3 (-1)^n + 9), {n, 0, 20}] (* Bruno Berselli, May 16 2013 *)
LinearRecurrence[{16, 16, -256}, {1, 10, 76}, 20] (* Bruno Berselli, May 17 2013 *)
CoefficientList[Series[(1 - 6 x - 100 x^2) / ((1 - 4 x) (1 + 4 x) (1 - 16 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Sep 04 2013 *)

a(n) = 16*a(n-1) + 16*a(n-2) - (16^2)*a(n-3) with n>2, a(0)=1, a(1)=10, a(2)=76.
a(n) = 2^(2n-3)*(2^(2n+1)-3*(-1)^n+9).
G.f.: (1-6*x-100*x^2)/((1-4*x)*(1+4*x)*(1-16*x)). [Bruno Berselli, May 16 2013]

More terms from Vincenzo Librandi, Sep 04 2013

A283432 Triangle read by rows: T(n,m) is the number of pattern classes in the (n,m)-rectangular grid with 3 colors and n>=m, 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, 3, 1, 6, 27, 1, 18, 216, 5346, 1, 45, 1701, 134865, 10766601, 1, 135, 15066, 3608550, 871858485, 211829725395, 1, 378, 133407, 96997824, 70607782701, 51472887053238, 37523659114815147, 1, 1134, 1198476, 2616461190, 5719211266905, 12507889858389450, 27354747358715650524, 59824832319304600777362

Offset: 0

María Merino, Imanol Unanue, Yosu Yurramendi, May 15 2017

Computed using Burnside's orbit-counting lemma.

			Triangle begins:
===========================================================
n\ m |   0  1     2      3         4           5
-----|-----------------------------------------------------
0    |   1
1    |   1  3
2    |   1  6     27
3    |   1  18    216    5346
4    |   1  45    1701   134865    10766601
5    |   1  135   15066  3608550   871858485   211829725395
...

María Merino, Rows n=0..46 of triangle, flattened
M. Merino and I. Unanue, Counting squared grid patterns with Pólya Theory, EKAIA, 34 (2018), 289-316 (in Basque).

Cf. A225910.

Table[Which[AllTrue[{n,m},EvenQ],(3^(m n)+3 3^((m n)/2))/4,EvenQ[ n]&&OddQ[m],(3^(m n)+3^((m n+n)/2)+2 3^((m n)/2))/4,OddQ[n]&&EvenQ[ m],(3^(m n)+3^((m n+m)/2)+2 3^((m n)/2))/4,True,(3^(m n)+3^((m n+n)/2)+3^((m n+m)/2)+3^((m n+1)/2))/4],{n,0,10},{m,0,n}]//Flatten (* Harvey P. Dale, Mar 29 2023 *)

For even n and m: T(n,m) = (3^(m*n) + 3*3^(m*n/2))/4;
for even n and odd m: T(n,m) = (3^(m*n) + 3^((m*n+n)/2) + 2*3^(m*n/2))/4;
for odd n and even m: T(n,m) = (3^(m*n) + 3^((m*n+m)/2) + 2*3^(m*n/2))/4;
for odd n and m: T(n,m) = (3^(m*n) + 3^((m*n+n)/2) + 3^((m*n+m)/2) + 3^((m*n+1)/2))/4.

A225829 Number of binary pattern classes in the (5,n)-rectangular grid: 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, 20, 288, 8640, 263680, 8407040, 268517376, 8590786560, 274882625536, 8796137062400, 281475261923328, 9007201737768960, 288230393868451840, 9223372185031147520, 295147906296044322816, 9444732974878980833280, 302231454974575793668096, 9671406557490978467348480

Offset: 0

Yosu Yurramendi, May 16 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..600
Index entries for linear recurrences with constant coefficients, signature (40,-224,-1280,8192).

A005418 is the number of binary pattern classes in the (1,n)-rectangular grid.
A225826 to A225834  are the numbers of binary pattern classes in the (m,n)-rectangular grid, 1 < m < 11 .
A225910 is the table of (m,n)-rectangular grids.

I:=[1, 20, 288, 8640]; [n le 4 select I[n] else 40*Self(n-1)-224*Self(n-2)-1280*Self(n-3)+8192*Self(n-4): n in [1..25]]; // Vincenzo Librandi, Sep 04 2013

LinearRecurrence[{40,-224,-1280,8192}, {1, 20, 288, 8640}, 20] (* Bruno Berselli, May 17 2013 *)
CoefficientList[Series[(1 - 20 x - 288 x^2 + 2880 x^3) / ((1 - 8 x) (1 - 32 x) (1 - 32 x^2)), {x, 0, 30}], x] (* Vincenzo Librandi, Sep 04 2013 *)

a(n) = 32*a(n-1) + 32*a(n-2) - 1024*a(n-3)- 2^(3n - 3)*3 with n>2, a(0)=1, a(1)=20, a(2)=288.
a(n) = 2^(5n/2-1)*(2^(5n/2-1) + 2^(n/2-1) + 1) if n is even,
a(n) = 2^((5n-1)/2-1)*(2^((5n-1)/2) + 2^((n-1)/2) + 5) if n is odd.
G.f.: (1-20*x-288*x^2+2880*x^3)/((1-8*x)*(1-32*x)*(1-32*x^2)). [Bruno Berselli, May 17 2013]

A283433 Triangle read by rows: T(n,m) is the number of pattern classes in the (n,m)-rectangular grid with 4 colors and n>=m, 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, 1, 10, 76, 1, 40, 1120, 67840, 1, 136, 16576, 4212736, 1073790976, 1, 544, 263680, 268779520, 274882625536, 281475530358784, 1, 2080, 4197376, 17184194560, 70368756760576, 288230393868451840, 1180591620768950910976

Offset: 0

María Merino, Imanol Unanue, Yosu Yurramendi, May 15 2017

Computed using Burnside's orbit-counting lemma.

			Triangle begins:
=======================================================================
n\m |  0   1       2        3           4               5
----|------------------------------------------------------------------
0   |  1
1   |  1    4
2   |  1    10     76
3   |  1    40     1120     67840
4   |  1    136    16576    4212736     1073790976
5   |  1    544    263680   268779520   274882625536    281475530358784
...

María Merino, Rows n=0..41 of triangle, flattened
M. Merino and I. Unanue, Counting squared grid patterns with Pólya Theory, EKAIA, 34 (2018), 289-316 (in Basque).

Cf. A225910, A283432.

For even n and m: T(n,m) = (4^(m*n) + 3*4^(m*n/2))/4;
for even n and odd m: T(n,m) = (4^(m*n) + 4^((m*n+n)/2) + 2*4^(m*n/2))/4;
for odd n and even m: T(n,m) = (4^(m*n) + 4^((m*n+m)/2) + 2*4^(m*n/2))/4;
for odd n and m: T(n,m) = (4^(m*n) + 4^((m*n+n)/2) + 4^((m*n+m)/2) + 4^((m*n+1)/2))/4.

A368218 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k grid up to horizontal and vertical reflections by a tile that is fixed under horizontal reflection only.

Original entry on oeis.org

1, 3, 2, 4, 7, 3, 10, 20, 24, 6, 16, 76, 144, 76, 10, 36, 272, 1120, 1056, 288, 20, 64, 1072, 8448, 16576, 8320, 1072, 36, 136, 4160, 66816, 262656, 263680, 65792, 4224, 72, 256, 16576, 528384, 4197376, 8396800, 4197376, 525312, 16576, 136

Offset: 1

Peter Kagey, Dec 18 2023

			Table begins:
  n\k |  1    2     3       4         5           6
  ----+--------------------------------------------
    1 |  1    3     4      10        16          36
    2 |  2    7    20      76       272        1072
    3 |  3   24   144    1120      8448       66816
    4 |  6   76  1056   16576    262656     4197376
    5 | 10  288  8320  263680   8396800   268517376
    6 | 20 1072 65792 4197376 268451840 17180065792

Peter Kagey, Illustration of T(3,2)=24
Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv: 2311.13072 [math.CO], 2023. See also J. Int. Seq., (2024) Vol. 27, Art. No. 24.6.1, pp. A-1, A-3.

Cf. A225910, A367524, A368219, A368220, A368221.

A368218[n_, m_] := 2^(n*m/2 - 2)*(2^(n*m/2) + Boole[EvenQ[n*m]] + Boole[EvenQ[m]] + If[EvenQ[n], 1, 2^(m/2)])

A225830 Number of binary pattern classes in the (6,n)-rectangular grid: 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, 36, 1072, 66816, 4197376, 268517376, 17180065792, 1099516870656, 70368756760576, 4503599962914816, 288230376957018112, 18446744095184388096, 1180591620768950910976, 75557863727288712953856, 4835703278461815233708032, 309485009821433029655003136, 19807040628566295504618520576, 1267650600228235030996237418496

Offset: 0

Yosu Yurramendi, May 16 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (64,64,-4096).

A005418 is the number of binary pattern classes in the (1,n)-rectangular grid.
A225826 to A225834  are the numbers of binary pattern classes in the (m,n)-rectangular grid, 1 < m < 11 .
A225910 is the table of (m,n)-rectangular grids.

I:=[1,36,1072]; [n le 3 select I[n] else 64*Self(n-1)+64*Self(n-2)-4096*Self(n-3): n in [1..25]]; // Vincenzo Librandi, Sep 04 2013

LinearRecurrence[{64, 64, -4096}, {1, 36, 1072}, 20] (* Bruno Berselli, May 17 2013 *)
CoefficientList[Series[(1 - 28 x - 1296 x^2) / ((1 - 8 x) (1 + 8 x) (1 - 64 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Sep 04 2013 *)

a(n) = 64*a(n-1) + 64*a(n-2) - (64^2)*a(n-3) with n>2, a(0)=1, a(1)=36, a(2)=1072.
a(n) = 2^(3n-3)*(2^(3n+1)-(2^3-1)*(-1)^n+2^3+5) = 8^(n-1)*(2^(3n+1)-7*(-1)^n+13).
G.f.: (1-28*x-1296*x^2)/((1-8*x)*(1+8*x)*(1-64*x)).

A283434 Triangle read by rows: T(n,m) is the number of pattern classes in the (n,m)-rectangular grid with 5 colors and n>=m, 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, 1, 15, 175, 1, 75, 4125, 496875, 1, 325, 98125, 61140625, 38147265625, 1, 1625, 2446875, 7632421875, 23841923828125, 74505821533203125, 1, 7875, 61046875, 953736328125, 14901161376953125, 232830644622802734375, 3637978807094573974609375

Offset: 0

María Merino, Imanol Unanue, Yosu Yurramendi, May 15 2017

Computed using Burnside's orbit-counting lemma.

			Triangle begins:
============================================================================
n\m |   0   1      2         3            4                5
----|-----------------------------------------------------------------------
0   |   1
1   |   1   5
2   |   1   15     175
3   |   1   75     4125      496875
4   |   1   325    98125     61140625     38147265625
5   |   1   1625   2446875   7632421875   23841923828125   74505821533203125
...

María Merino, Rows n=0..38 of triangle, flattened
M. Merino and I. Unanue, Counting squared grid patterns with Pólya Theory, EKAIA, 34 (2018), 289-316 (in Basque).

Cf. A225910, A283432, A283433.

For even n and m: T(n,m) = (5^(m*n) + 3*5^(m*n/2))/4;
for even n and odd m: T(n,m) = (5^(m*n) + 5^((m*n+n)/2) + 2*5^(m*n/2))/4;
for odd n and even m: T(n,m) = (5^(m*n) + 5^((m*n+m)/2) + 2*5^(m*n/2))/4;
for odd n and m: T(n,m) = (5^(m*n) + 5^((m*n+n)/2) + 5^((m*n+m)/2) + 5^((m*n+1)/2))/4.

cp's OEIS Frontend

A225826 Number of binary pattern classes in the (2,n)-rectangular grid: 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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A225834 Number of binary pattern classes in the (10,n)-rectangular grid: 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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A225827 Number of binary pattern classes in the (3,n)-rectangular grid: 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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A225828 Number of binary pattern classes in the (4,n)-rectangular grid: 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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A283432 Triangle read by rows: T(n,m) is the number of pattern classes in the (n,m)-rectangular grid with 3 colors and n>=m, 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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A225829 Number of binary pattern classes in the (5,n)-rectangular grid: 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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A283433 Triangle read by rows: T(n,m) is the number of pattern classes in the (n,m)-rectangular grid with 4 colors and n>=m, 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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A368218 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k grid up to horizontal and vertical reflections by a tile that is fixed under horizontal reflection only.

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