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))])

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])

A368305 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k torus up to horizontal reflections by two tiles that are both fixed under horizontal reflection.

Original entry on oeis.org

2, 3, 3, 4, 7, 4, 6, 14, 13, 6, 8, 40, 44, 34, 8, 14, 108, 218, 226, 78, 13, 20, 362, 1200, 2386, 1184, 237, 18, 36, 1182, 7700, 27936, 26892, 7700, 687, 30, 60, 4150, 51112, 361244, 674384, 354680, 50628, 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        14
   2 |  3   7   14     40      108       362
   3 |  4  13   44    218     1200      7700
   4 |  6  34  226   2386    27936    361244
   5 |  8  78 1184  26892   674384  17920876
   6 | 13 237 7700 354680 17950356 955180432

Links

Crossrefs

Cf. A368221, A368258, A368302, A368306.

Programs

  • Mathematica
    A368305[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/#))&]/2, DivisorSum[m, EulerPhi[#](2^((n - 1)*m/LCM[2, #])*2^(m/#))&]])
Showing 1-3 of 3 results.

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

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