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.

A368302 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 horizontal reflections but not vertical reflections.

Original entry on oeis.org

1, 2, 2, 2, 5, 2, 4, 9, 8, 4, 4, 26, 22, 22, 4, 9, 62, 120, 126, 44, 8, 10, 205, 600, 1267, 592, 135, 9, 22, 623, 3936, 14164, 13600, 3936, 362, 18, 30, 2171, 25556, 181782, 337192, 178366, 25314, 1211, 23, 62, 7429, 177678, 2437726, 8965354, 8980642, 2404372, 176998, 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         9
   2 | 2   5    9     26      62       205
   3 | 2   8   22    120     600      3936
   4 | 4  22  126   1267   14164    181782
   5 | 4  44  592  13600  337192   8965354
   6 | 8 135 3936 178366 8980642 477655760

Links

Crossrefs

Cf. A222188, A368218, A368254, A368303, A368304, A368305.

Programs

  • Mathematica
    A368302[n_, m_] := 1/(4*n*m) (DivisorSum[n, Function[d, DivisorSum[m, EulerPhi[#] EulerPhi[d] 2^(m*n/LCM[#, d]) &]]] + n*If[EvenQ[n], 1/2*DivisorSum[m, EulerPhi[#] (2^(n*m/LCM[2, #]) + 2^((n - 2)*m/LCM[2, #])*2^(2 m/#)) &], DivisorSum[m, EulerPhi[#] (2^((n - 1)*m/LCM[2, #])*2^(m/#)) &]] + m*If[EvenQ[m], 1/2*DivisorSum[n, EulerPhi[#] (2^(n*m/LCM[2, #]) + 2^(m*n/#)*Boole[EvenQ[#]]) &], DivisorSum[n, EulerPhi[#]*2^(m*n/#) &, EvenQ]] + n*m*2^(n*m/2)*Which[EvenQ[n] && EvenQ[m], 3/4, OddQ[n*m], 0, OddQ[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])

A368306 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k torus up to horizontal reflections by a tile that is not fixed under horizontal reflection.

Original entry on oeis.org

1, 2, 2, 2, 5, 2, 4, 8, 9, 4, 4, 24, 32, 26, 4, 8, 56, 186, 182, 62, 9, 10, 190, 1096, 2130, 1096, 205, 10, 20, 596, 7356, 26296, 26380, 7356, 623, 22, 30, 2102, 49940, 350316, 671104, 350584, 49940, 2171, 30
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     24       56       190
   3 | 2   9   32    186     1096      7356
   4 | 4  26  182   2130    26296    350316
   5 | 4  62 1096  26380   671104  17899020
   6 | 9 205 7356 350584 17897924 954481360

Links

Crossrefs

Cf. A368222, A368259, A368304, A368305, A368308, A184271.

Programs

  • Mathematica
    A368306[n_, m_] := 1/(2*n*m)*(DivisorSum[n, Function[d, DivisorSum[m, EulerPhi[#] EulerPhi[d] 2^(m*n/LCM[#, d]) &]]] + n*If[EvenQ[n], DivisorSum[m, EulerPhi[#] (2^(n*m/LCM[2, #]) + 2^((n - 2)*m/LCM[2, #])*4^(m/#)*Boole[EvenQ[#]]) &]/2, DivisorSum[m, EulerPhi[#]*2^(n*m/#) &, EvenQ]])

A368308 Table read by antidiagonals: T(n,k) is the number of tilings of the n X k torus up to 180-degree rotation by a tile that is not fixed under 180-degree rotation.

Original entry on oeis.org

1, 2, 2, 2, 5, 2, 4, 9, 9, 4, 4, 26, 32, 26, 4, 9, 62, 192, 192, 62, 9, 10, 205, 1096, 2174, 1096, 205, 10, 22, 623, 7440, 26500, 26500, 7440, 623, 22, 30, 2171, 49940, 351336, 671104, 351336, 49940, 2171, 30
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         9
   2 | 2   5    9     26       62       205
   3 | 2   9   32    192     1096      7440
   4 | 4  26  192   2174    26500    351336
   5 | 4  62 1096  26500   671104  17904476
   6 | 9 205 7440 351336 17904476 954546880

Links

Crossrefs

Cf. A184271, A368224, A368263, A368304, A368306, A368307.

Programs

  • Mathematica
    A368308[n_, m_] := 1/(2*n*m)*(DivisorSum[n, Function[d, DivisorSum[m, EulerPhi[#] EulerPhi[d] 2^(m*n/LCM[#, d]) &]]] + n*m*2^(n*m/2)*Which[OddQ[n*m], 0, OddQ[n + m], 1/2, True, 3/4])
Showing 1-4 of 4 results.

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