cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A368220 Table read by antidiagonals: T(n,k) is the number of tilings of the n X k grid up to horizontal and vertical reflections by an asymmetric tile.

Original entry on oeis.org

1, 6, 6, 16, 76, 16, 72, 1056, 1056, 72, 256, 16576, 65536, 16576, 256, 1056, 262656, 4196352, 4196352, 262656, 1056, 4096, 4197376, 268435456, 1073790976, 268435456, 4197376, 4096, 16512, 67117056, 17180000256, 274878431232, 274878431232, 17180000256, 67117056, 16512
Offset: 1

Views

Author

Peter Kagey, Dec 18 2023

Keywords

Examples

			Table begins:
  n\k |    1       2           3              4                  5
  ----+-----------------------------------------------------------
    1 |    1       6          16             72                256
    2 |    6      76        1056          16576             262656
    3 |   16    1056       65536        4196352          268435456
    4 |   72   16576     4196352     1073790976       274878431232
    5 |  256  262656   268435456   274878431232    281474976710656
    6 | 1056 4197376 17180000256 70368756760576 288230376688582656

Links

Crossrefs

Cf. A117401, A225910, A368218, A368219, A368222, A368224.

Programs

  • Mathematica
    A368220[n_, m_] := 2^(n*m - 2)*(2^(n*m) + Boole[EvenQ[n*m]] + Boole[EvenQ[n]] + Boole[EvenQ[m]])

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

Views

Author

Yosu Yurramendi, May 16 2013

Keywords

Links

Crossrefs

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.

Programs

  • Magma
    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
  • Mathematica
    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 *)

Formula

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]

A286893 Triangle read by rows: T(n,m) is the number of pattern classes in the (n,m)-rectangular grid with 6 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, 6, 1, 21, 351, 1, 126, 12096, 2544696, 1, 666, 420876, 544638816, 705278736576, 1, 3996, 15132096, 117564302016, 914040184444416, 7107572245840091136, 1, 23436, 544230576, 25390538401536, 1184595336212990976, 55268479955808421134336, 2578606199622710056510488576
Offset: 0

Views

Author

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

Keywords

Comments

Computed using Burnside's orbit-counting lemma.

Examples

			Triangle begins:
============================================================================
n\m |  0  1     2         3             4                5
----|-----------------------------------------------------------------------
0   |  1
1   |  1  6
2   |  1  21    351
3   |  1  126   12096     2544696
4   |  1  666   420876    544638816     705278736576
5   |  1  3996  15132096  117564302016  914040184444416  7107572245840091136
...

Links

Crossrefs

Cf. A225910, A283432, A283433, A283434.

Formula

For even n and m: T(n,m) = (6^(m*n) + 3*6^(m*n/2))/4;
for even n and odd m: T(n,m) = (6^(m*n) + 6^((m*n+n)/2) + 2*6^(m*n/2))/4;
for odd n and even m: T(n,m) = (6^(m*n) + 6^((m*n+m)/2) + 2*6^(m*n/2))/4;
for odd n and m: T(n,m) = (6^(m*n) + 6^((m*n+n)/2) + 6^((m*n+m)/2) + 6^((m*n+1)/2))/4.

A368219 Table read by 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 180-degree rotation but not horizontal or vertical reflection.

Original entry on oeis.org

1, 2, 2, 3, 7, 3, 6, 20, 20, 6, 10, 76, 136, 76, 10, 20, 272, 1056, 1056, 272, 20, 36, 1072, 8256, 16576, 8256, 1072, 36, 72, 4160, 65792, 262656, 262656, 65792, 4160, 72, 136, 16576, 524800, 4197376, 8390656, 4197376, 524800, 16576, 136
Offset: 1

Views

Author

Peter Kagey, Dec 18 2023

Keywords

Examples

			Table begins:
  n\k |  1    2     3       4         5           6
  ----+--------------------------------------------
    1 |  1    2     3       6        10          20
    2 |  2    7    20      76       272        1072
    3 |  3   20   136    1056      8256       65792
    4 |  6   76  1056   16576    262656     4197376
    5 | 10  272  8256  262656   8390656   268451840
    6 | 20 1072 65792 4197376 268451840 17180065792

Links

Crossrefs

Cf. A225910, A368218, A368220, A368223.

Programs

  • Mathematica
    A368219[n_, m_] := 2^(n*m/2 - 2)*(2^(n*m/2) + If[EvenQ[n*m], 1, Sqrt[2]] + Boole[EvenQ[n]] + Boole[EvenQ[m]])

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

Views

Author

Yosu Yurramendi, May 16 2013

Keywords

Links

Crossrefs

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.

Programs

  • Magma
    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
  • Mathematica
    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 *)

Formula

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)).

A286895 Triangle read by rows: T(n,m) is the number of pattern classes in the (n,m)-rectangular grid with 7 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, 7, 1, 28, 637, 1, 196, 30184, 10151428, 1, 1225, 1443001, 3461821825, 8308236966001, 1, 8575, 70656628, 1186972525900, 19948070175962425, 335267157313994232775, 1, 58996, 3460410037, 407106879976216, 47895307855522569001, 5634835073082541702198396, 662932711464914589254954278237
Offset: 0

Views

Author

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

Keywords

Comments

Computed using Burnside's orbit-counting lemma.

Examples

			Triangle begins:
============================================================================
n\m |  0  1    2        3             4                 5
----|-----------------------------------------------------------------------
0   |  1
1   |  1  7
2   |  1  28   637
3   |  1  196  30184    10151428
4   |  1  1225 1443001  3461821825    8308236966001
5   |  1  8575 70656628 1186972525900 19948070175962425 335267157313994232775
...

Links

Crossrefs

Cf. A225910, A283432, A283433, A283434, A286893.

Formula

For even n and m: T(n,m) = (7^(m*n) + 3*7^(m*n/2))/4;
for even n and odd m: T(n,m) = (7^(m*n) + 7^((m*n+n)/2) + 2*7^(m*n/2))/4;
for odd n and even m: T(n,m) = (7^(m*n) + 7^((m*n+m)/2) + 2*7^(m*n/2))/4;
for odd n and m: T(n,m) = (7^(m*n) + 7^((m*n+n)/2) + 7^((m*n+m)/2) + 7^((m*n+1)/2))/4.

A286919 Triangle read by rows: T(n,m) is the number of pattern classes in the (n,m)-rectangular grid with 8 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, 8, 1, 36, 1072, 1, 288, 66816, 33693696, 1, 2080, 4197376, 17184194560, 70368756760576, 1, 16640, 268517376, 8796399206400, 288230393868451840, 9444732983468915425280, 1, 131328, 17180065792, 4503616874348544, 1180591620768950910976, 309485009825866260538195968, 81129638414606695206587887255552
Offset: 0

Views

Author

María Merino, Imanol Unanue, Yosu Yurramendi, May 16 2017

Keywords

Comments

Computed using Burnsides orbit-counting lemma.

Examples

			Triangle begins:
========================================================
n\m |   0   1      2        3             4
----|---------------------------------------------------
0   |   1
1   |   1   8
2   |   1   36     1072
3   |   1   288    66816    33693696
4   |   1   2080   4197376  17184194560   70368756760576
...

Links

Crossrefs

Cf. A225910, A283432, A283433, A283434, A286893, A286895.

Formula

For even n and m: T(n,m) = (8^(m*n) + 3*8^(m*n/2))/4;
for even n and odd m: T(n,m) = (8^(m*n) + 8^((m*n+n)/2) + 2*8^(m*n/2))/4;
for odd n and even m: T(n,m) = (8^(m*n) + 8^((m*n+m)/2) + 2*8^(m*n/2))/4;
for odd n and m: T(n,m) = (8^(m*n) + 8^((m*n+n)/2) + 8^((m*n+m)/2) + 8^((m*n+1)/2))/4.

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

Views

Author

Peter Kagey, Dec 19 2023

Keywords

Examples

			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

Links

Crossrefs

Cf. A222188, A225910.
Cf. A005418 (n=1), A225826 (n=2), A000029 (k=1), A222187 (k=2).

Programs

  • Mathematica
    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)])

A225833 Number of binary pattern classes in the (9,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, 272, 66048, 33632256, 17180262400, 8796137062400, 4503599962914816, 2305843036057239552, 1180591621026648948736, 604462909825456529211392, 309485009821644135887536128, 158456325028542467460946722816
Offset: 0

Views

Author

Yosu Yurramendi, May 16 2013

Keywords

Links

Crossrefs

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.

Programs

  • Magma
    [2^(5*n-2)+2^(9*n-2)+(34-(17-Sqrt(2))*(1+(-1)^n))*Sqrt(2)^(9*n-5): n in [0..16]]; // Vincenzo Librandi, Sep 04 2013
  • Mathematica
    LinearRecurrence[{544, -15872, -278528, 8388608}, {1, 272, 66048, 33632256}, 20] (* Bruno Berselli, May 17 2013 *)
CoefficientList[Series[(1 - 272 x - 66048 x^2 + 2297856 x^3) / ((1 - 32 x) (1 - 512 x) (1 - 512 x^2)), {x, 0, 30}], x] (* Vincenzo Librandi, Sep 04 2013 *)

Formula

a(n) = 2^9*a(n-1) + 2^9*a(n-2) - (2^9)^2*a(n-3) - 2^(((9+1)/2)*n - 3)*(2^((9-1)/2)-1) with n>2, a(0)=1, a(1)=272, a(2)=66048.
a(n) = 2^(9n/2-1)*(2^(9n/2-1) + 2^(n/2-1) + 1) if n is even,
a(n) = 2^((9n-1)/2-1)*(2^((9n-1)/2) + 2^((n-1)/2) + 2^((9-1)/2) + 1) if n is odd.
G.f.: (1-272*x-66048*x^2+2297856*x^3)/((1-32*x)*(1-512*x)*(1-512*x^2)). [Bruno Berselli, May 17 2013]
a(n) = 2^(5n-2)+2^(9n-2)+(34-(17-sqrt(2))*(1+(-1)^n))*sqrt(2)^(9n-5). [Bruno Berselli, May 17 2013]

A286920 Triangle read by rows: T(n,m) is the number of pattern classes in the (n,m)-rectangular grid with 9 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, 9, 1, 45, 1701, 1, 405, 134865, 97135605, 1, 3321, 10766601, 70618411521, 463255079498001, 1, 29889, 871858485, 51473762336565, 3039416437115008521, 179474497026544179696969, 1, 266085, 70607782701, 37523729625344145, 19941610769429949618201, 10597789568841677482963905405, 5632099886234793715531013441442501
Offset: 0

Views

Author

María Merino, Imanol Unanue, Yosu Yurramendi, May 16 2017

Keywords

Comments

Computed using Burnsides orbit-counting lemma.

Examples

			Triangle begins:
==========================================================
n\m |   0   1     2         3              4
----|-----------------------------------------------------
0   |   1
1   |   1   9
2   |   1   45    1701
3   |   1   405   134865    97135605
4   |   1   3321  10766601  70618411521    463255079498001
...

Links

Crossrefs

Cf. A225910, A283432, A283433, A283434, A286893, A286895, A286919.

Formula

For even n and m: T(n,m) = (9^(m*n) + 3*9^(m*n/2))/4;
for even n and odd m: T(n,m) = (9^(m*n) + 9^((m*n+n)/2) + 2*9^(m*n/2))/4;
for odd n and even m: T(n,m) = (9^(m*n) + 9^((m*n+m)/2) + 2*9^(m*n/2))/4;
for odd n and m: T(n,m) = (9^(m*n) + 9^((m*n+n)/2) + 9^((m*n+m)/2) + 9^((m*n+1)/2))/4.
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