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.

Showing 1-4 of 4 results.

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

Views

Author

Peter Kagey, Dec 18 2023

Keywords

Examples

			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

Links

Crossrefs

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

Programs

  • Mathematica
    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)])

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]])

A368223 Table read by antidiagonals: T(n,k) is the number of tilings of the n X k grid up to 180-degree rotation by two tiles that are each fixed under 180-degree rotation.

Original entry on oeis.org

2, 3, 3, 6, 10, 6, 10, 36, 36, 10, 20, 136, 272, 136, 20, 36, 528, 2080, 2080, 528, 36, 72, 2080, 16512, 32896, 16512, 2080, 72, 136, 8256, 131328, 524800, 524800, 131328, 8256, 136, 272, 32896, 1049600, 8390656, 16781312, 8390656, 1049600, 32896, 272
Offset: 1

Views

Author

Peter Kagey, Dec 18 2023

Keywords

Examples

			Table begins:
  n\k|  1    2      3       4         5           6
  ---+---------------------------------------------
   1 |  2    3      6      10        20          36
   2 |  3   10     36     136       528        2080
   3 |  6   36    272    2080     16512      131328
   4 | 10  136   2080   32896    524800     8390656
   5 | 20  528  16512  524800  16781312   536887296
   6 | 36 2080 131328 8390656 536887296 34359869440

Links

Crossrefs

Cf. A368219, A368224.

Programs

  • Mathematica
    A368223[n_, m_] := 1/2 (2^(n*m) + If[EvenQ[n*m], 2^(n*m/2), 2^((n*m + 1)/2)])

A368303 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k torus up to horizontal and vertical reflections by a tile that is fixed under 180-degree rotations but not horizontal or vertical reflections.

Original entry on oeis.org

1, 2, 2, 2, 5, 2, 4, 8, 8, 4, 4, 22, 24, 22, 4, 8, 44, 120, 120, 44, 8, 9, 135, 612, 1203, 612, 135, 9, 18, 362, 3892, 13600, 13600, 3892, 362, 18, 23, 1211, 25482, 177342, 337600, 177342, 25482, 1211, 23, 44, 3914, 176654, 2404372, 8962618, 8962618, 2404372, 176654, 3914, 44
Offset: 1

Views

Author

Peter Kagey, Dec 21 2023

Keywords

Examples

			Table begins:
  n\k| 1   2    3      4       5         6
  ---+------------------------------------
   1 | 1   2    2      4       4         8
   2 | 2   5    8     22      44       135
   3 | 2   8   24    120     612      3892
   4 | 4  22  120   1203   13600    177342
   5 | 4  44  612  13600  337600   8962618
   6 | 8 135 3892 177342 8962618 477371760

Links

Crossrefs

Cf. A222188, A368219, A368256, A368302, A368304, A368307.

Programs

  • Mathematica
    A368303[n_, m_]:=1/(4*n*m)*(DivisorSum[n, Function[d, DivisorSum[m, Function[c, EulerPhi[c]EulerPhi[d]2^(m*n/LCM[c, d])]]]] + If[EvenQ[n], n/2*DivisorSum[m, EulerPhi[#](2^(n*m/LCM[2, #]) + 2^((n - 2)*m/LCM[2, #])*2^(2m/#)*Boole[EvenQ[#]])&], n*DivisorSum[m, EulerPhi[#](2^(n*m/#))&, EvenQ]] + If[EvenQ[m], m/2*DivisorSum[n, EulerPhi[#](2^(n*m/LCM[2, #]) + 2^((m - 2)*n/LCM[2, #])*2^(2n/#)*Boole[EvenQ[#]])&], m*DivisorSum[n, EulerPhi[#](2^(m*n/#))&, EvenQ]] + n*m*2^((n*m)/2)*Which[OddQ[n*m], Sqrt[2], OddQ[n + m], 3/2, True, 7/4])
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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