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 1-3 of 3 results.

A368304 Table read by antidiagonals: T(n,k) is the number of tilings of the n X k torus up to horizontal and vertical reflections by an asymmetric tile.

Original entry on oeis.org

1, 4, 4, 6, 28, 6, 23, 194, 194, 23, 52, 2196, 7296, 2196, 52, 194, 26524, 350573, 350573, 26524, 194, 586, 351588, 17895736, 67136624, 17895736, 351588, 586, 2131, 4798174, 954495904, 13744131446, 13744131446, 954495904, 4798174, 2131
Offset: 1

Views

Author

Peter Kagey, Dec 21 2023

Keywords

Examples

			Table begins:
  n\k|   1      2         3             4                5
  ---+----------------------------------------------------
   1 |   1      4         6            23               52
   2 |   4     28       194          2196            26524
   3 |   6    194      7296        350573         17895736
   4 |  23   2196    350573      67136624      13744131446
   5 |  52  26524  17895736   13744131446   11258999068672
   6 | 194 351588 954495904 2932037300956 9607679419823148

Links

Crossrefs

Cf. A222188, A368220, A368257, A368302, A368303, A368306, A368308, A184271.

Programs

  • Mathematica
    A368304[n_,m_]:=1/(4*n*m) (DivisorSum[n, Function[d,DivisorSum[m,Function[c,EulerPhi[c]EulerPhi[d]4^(m*n/LCM[c,d])]]]]+If[EvenQ[n],n/2*DivisorSum[m, EulerPhi[#](4^(n*m/LCM[2,#])+4^((n-2)*m/LCM[2,#])*4^(2m/#)*Boole[EvenQ[#]])&],n*DivisorSum[m,EulerPhi[#](4^(n*m/#))&,EvenQ]]+If[EvenQ[m], m/2*DivisorSum[n,EulerPhi[#](4^(n*m/LCM[2,#])+4^((m-2)*n/LCM[2,#])*4^(2n/#)*Boole[EvenQ[#]])&],m*DivisorSum[n, EulerPhi[#](4^(m*n/#))&,EvenQ]]+Which[EvenQ[n]&&EvenQ[m],(n*m)/4 (3*2^(n*m)),OddQ[n*m],0,OddQ[n+m],(n*m)/2 (2^(n*m))])

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])

A368307 Table read by antidiagonals: T(n,k) is the number of tilings of the n X k torus up to 180-degree rotation by two tiles that are both fixed under 180-degree rotation.

Original entry on oeis.org

2, 3, 3, 4, 7, 4, 6, 13, 13, 6, 8, 34, 48, 34, 8, 13, 78, 224, 224, 78, 13, 18, 237, 1224, 2302, 1224, 237, 18, 30, 687, 7696, 27012, 27012, 7696, 687, 30, 46, 2299, 50964, 353384, 675200, 353384, 50964, 2299, 46
Offset: 1

Views

Author

Peter Kagey, Dec 21 2023

Keywords

Examples

			Table begins:
  n\k|  1   2    3      4        5         6
  ---+--------------------------------------
   1 |  2   3    4      6        8        13
   2 |  3   7   13     34       78       237
   3 |  4  13   48    224     1224      7696
   4 |  6  34  224   2302    27012    353384
   5 |  8  78 1224  27012   675200  17920860
   6 | 13 237 7696 353384 17920860 954677952

Links

Crossrefs

Cf. A368223, A368262, A368303, A368308.

Programs

  • Mathematica
    A368307[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], Sqrt[2], OddQ[n + m], 3/2, True, 7/4])
Showing 1-3 of 3 results.

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

Content is available under The OEIS End-User License Agreement.