A225826 -id:A225826 - cp's OEIS frontend

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.

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

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

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]

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)).

A225831 Number of binary pattern classes in the (7,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, 72, 4224, 529920, 67133440, 8590786560, 1099516870656, 140737630961664, 18014399717441536, 2305843036057239552, 295147905471410601984, 37778931868592158801920, 4835703278531084466257920, 618970019643974367030804480, 79228162514282633467030142976, 10141204801826143708548100521984, 1298074214633711554847439528656896, 166153499473114560494025562738655232

Offset: 0

Yosu Yurramendi, May 16 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (144,-1920,-18432,262144).

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:=[1,72,4224,529920]; [n le 4 select I[n] else 144*Self(n-1)-1920*Self(n-2)-18432*Self(n-3)+262144*Self(n-4): n in [1..20]]; // Vincenzo Librandi, Sep 04 2013

LinearRecurrence[{144, -1920, -18432, 262144}, {1, 72, 4224, 529920}, 20] (* Bruno Berselli, May 17 2013 *)
CoefficientList[Series[(1 - 72 x - 4224 x^2 + 78336 x^3) / ((1 - 16 x) (1 - 128 x) (1 - 128 x^2)), {x, 0, 30}], x] (* Vincenzo Librandi, Sep 04 2013 *)

a(n) = (2^7)*a(n-1) + (2^7)*a(n-2) - ((2^7)^2)*a(n-3) - 2^(4n-3)*7 with n>2, a(0)=1, a(1)=72, a(2)=4224.
a(n) = 2^(7n/2-1)*(2^(7n/2-1) + 2^(n/2-1) + 1) if n is even,
a(n) = 2^((7n-1)/2-1)*(2^((7n-1)/2) + 2^((n-1)/2) + 9) if n is odd.
G.f.: (1-72*x-4224*x^2+78336*x^3)/((1-16*x)*(1-128*x)*(1-128*x^2)). [Bruno Berselli, May 17 2013]

A225832 Number of binary pattern classes in the (8,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, 136, 16576, 4212736, 1073790976, 274882625536, 70368756760576, 18014399717441536, 4611686021648613376, 1180591621026648948736, 302231454904481927397376, 77371252455415432018395136, 19807040628566295504618520576

Offset: 0

Yosu Yurramendi, May 16 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (256,256,-65536).

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:=[1,136,16576]; [n le 3 select I[n] else 256*Self(n-1)+256*Self(n-2)-65536*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Sep 04 2013

CoefficientList[Series[(1 - 120 x - 18496 x^2) / ((1 - 16 x) (1 + 16 x) (1 - 256 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Sep 04 2013 *)

a(n) = 2^8*a(n-1) + 2^8*a(n-2) - (2^8)^2*a(n-3), with n>2, a(0)=1, a(1)=136, a(2)=16576.
a(n) = 2^(4n-3)*(2^(4n+1)-(2^4-1)*(-1)^n+2^4+5).
G.f.: (1-120*x-18496*x^2)/((1-16*x)*(1+16*x)*(1-256*x)).

A368253 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k cylinder up to horizontal and vertical reflections by two tiles that are fixed under these reflections.

Original entry on oeis.org

2, 3, 3, 6, 7, 4, 10, 24, 13, 6, 20, 76, 74, 34, 8, 36, 288, 430, 378, 78, 13, 72, 1072, 3100, 4756, 1884, 237, 18, 136, 4224, 23052, 70536, 53764, 11912, 687, 30, 272, 16576, 179736, 1083664, 1689608, 709316, 77022, 2299, 46

Offset: 1

Peter Kagey, Dec 19 2023

			Table begins:
  n\k |  1   2     3      4        5          6
  ----+----------------------------------------
    1 |  2   3     6     10       20         36
    2 |  3   7    24     76      288       1072
    3 |  4  13    74    430     3100      23052
    4 |  6  34   378   4756    70536    1083664
    5 |  8  78  1884  53764  1689608   53762472
    6 | 13 237 11912 709316 44900448 2865540112

Peter Kagey, Illustration of T(2,3)=24
Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv:2311.13072 [math.CO], 2023.

Cf. A222188, A225910.

Cf. A005418 (n=1), A225826 (n=2), A000029 (k=1), A222187 (k=2).

A368253[n_, m_] := 1/(4n)*(DivisorSum[n, Function[d, EulerPhi[d]*2^(n*m/d)]] + n*If[EvenQ[n], 1/2 (2^((n*m + 2 m)/2) + 2^(n*m/2)), 2^((n*m + m)/2)] + If[EvenQ[m], DivisorSum[n, Function[d, EulerPhi[d]*2^(n*m/LCM[d, 2])]], DivisorSum[n, Function[d, EulerPhi[d]*2^((n*m - n)/LCM[d, 2])*2^(n/d)]]] + n*Which[EvenQ[m], 2^(n*m/2), OddQ[m] && EvenQ[n], (3/2*2^(n*m/2)), OddQ[m] && OddQ[n], 2^((n*m + 1)/2)])

A132390 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 reflection or rotation of the other one.

Original entry on oeis.org

3, 6, 24, 76, 288, 1072, 4224, 16576, 66048, 262912, 1050624, 4197376, 16785408, 67121152, 268468224, 1073790976, 4295098368, 17180065792, 68720001024, 274878693376, 1099513724928, 4398049656832, 17592194433024, 70368756760576

Offset: 1

Yosu Yurramendi, Aug 26 2008

A005418 is the solution for the problem in the (1,n)-rectangular grid.
For n != 2, a(n) = 4^(n-1) + 2*A133572(n-1). - Jon E. Schoenfield, Aug 25 2009
A225826 is the same sequence, except a(2)=7. Here, 90-degree rotation is allowed, so a(2)=6. [Yosu Yurramendi, May 18 2013 - communicated by Jon E. Schoenfield]

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,4,-16).

Cf. A005418, A034851.

I:=[3,6,24,76,288]; [n le 5 select I[n] else 4*Self(n-1)+4*Self(n-2)-16*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Sep 04 2013

CoefficientList[Series[-(16 x^4 - 4 x^3 + 12 x^2 + 6 x - 3) / ((2 x - 1) (2 x + 1) (4 x - 1)), {x, 0, 30}], x] (* Vincenzo Librandi, Sep 04 2013 *)
LinearRecurrence[{4,4,-16},{3,6,24,76,288},30] (* Harvey P. Dale, Sep 22 2016 *)

For n != 2, a(n) = 4^(n-1) + 2^(n-2)*(3 + (n mod 2)). - Jon E. Schoenfield, Aug 25 2009
From Colin Barker, May 20 2013: (Start)
a(n) = 2^(-3+n)*(7 - (-1)^n + 2^(1+n)) for n > 2.
a(n) = 4*a(n-1) + 4*a(n-2) - 16*a(n-3), n >= 6.
G.f.: -x*(16*x^4 - 4*x^3 + 12*x^2 + 6*x - 3) / ((2*x-1)*(2*x+1)*(4*x-1)). (End)

More terms from Jon E. Schoenfield, Aug 25 2009, corrected Aug 30 2009

cp's OEIS Frontend

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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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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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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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.

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A225831 Number of binary pattern classes in the (7,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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A225832 Number of binary pattern classes in the (8,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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