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 1-10 of 10 results.

A233427 Number A(n,k) of tilings of a k X n rectangle using pentominoes of any shape; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 5, 0, 0, 5, 0, 1, 1, 0, 0, 56, 0, 56, 0, 0, 1, 1, 0, 0, 0, 501, 501, 0, 0, 0, 1, 1, 0, 0, 0, 0, 4006, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 27950, 27950, 0, 0, 0, 1, 1, 1, 0, 45, 0, 0, 214689, 0, 214689, 0, 0, 45, 0, 1
Offset: 0

Views

Author

Alois P. Heinz, Dec 09 2013

Keywords

Examples

			A(5,2) = A(2,5) = 5:
  ._________. ._________. ._________. ._________. ._________.
  |_________| | ._____| | | |_____. | |   ._|   | |   |_.   |
  |_________| |_|_______| |_______|_| |___|_____| |_____|___|.
Square array A(n,k) begins:
  1, 1,  1,    1,      1,         1,          1, ...
  1, 0,  0,    0,      0,         1,          0, ...
  1, 0,  0,    0,      0,         5,          0, ...
  1, 0,  0,    0,      0,        56,          0, ...
  1, 0,  0,    0,      0,       501,          0, ...
  1, 1,  5,   56,    501,      4006,      27950, ...
  1, 0,  0,    0,      0,     27950,          0, ...
  1, 0,  0,    0,      0,    214689,          0, ...
  1, 0,  0,    0,      0,   1696781,          0, ...
  1, 0,  0,    0,      0,  13205354,          0, ...
  1, 1, 45, 7670, 890989, 101698212, 7845888732, ...
  ...

Links

Crossrefs

Columns (or rows) include: A000012, A054318, A233428, A233429, A174249, A233430.
Cf. A099390, A233320, A230031, A246902, A247117, A278657.
Row sums of A247702, A247703, A247704, A247705, A247706, A247707, A247708, A247709, A247710, A247711, A247712, A247713 give A(n,5).

Formula

A(n,k) = 0 <=> n*k mod 5 > 0.

A278456 Number of tilings of a 5 X n rectangle using pentominoes of any shape and monominoes.

Original entry on oeis.org

1, 2, 50, 1954, 56864, 1532496, 42238426, 1178422563, 32890293494, 917103556607, 25552076570350, 711923354658732, 19838824712825618, 552851181380560869, 15406086995815163663, 429312063890812931103, 11963383230714027535776, 333377000620725693771782
Offset: 0

Views

Author

Alois P. Heinz, Nov 22 2016

Keywords

Examples

			a(1) = 2:
._.   ._.
|_|   | |
|_|   | |
|_|   | |
|_|   | |
|_|   |_|  .

Links

Crossrefs

Column k=5 of A278657.
Cf. A174249, A233427, A273477.

A278877 Number of tilings of a 6 X n rectangle using pentominoes of any shape and monominoes.

Original entry on oeis.org

1, 3, 155, 11914, 728857, 42238426, 2492016728, 147382229321, 8704392890599, 514207059584879, 30367424047335204, 1793195869965086337, 105893252956708204722, 6253339527216096201144, 369279138864066223011929, 21807048314051213226593407
Offset: 0

Views

Author

Alois P. Heinz, Nov 29 2016

Keywords

Examples

			a(1) = 3:
._.   ._.   ._.
|_|   | |   |_|
|_|   | |   | |
|_|   | |   | |
|_|   | |   | |
|_|   |_|   | |
|_|   |_|   |_|  .

Links

Crossrefs

Column k=6 of A278657.

A278964 Number of tilings of a 6 X n rectangle using pentominoes of shapes L, U, X and monominoes.

Original entry on oeis.org

1, 1, 31, 432, 9851, 150749, 2392788, 38885816, 646657365, 10599739324, 173226495187, 2835753730682, 46501906844608, 762113847626540, 12484929995911672, 204541850515528677, 3351522453919340317, 54915791725126323755, 899777273419073185968
Offset: 0

Views

Author

Alois P. Heinz, Dec 02 2016

Keywords

Examples

			a(2) = 31:
.___. .___. .___. .___. .___. .___. .___. .___. .___. .___. .___.
|_|_| | ._| | ._| |_|_| |_|_| |_|_| |_. | |_. | |_|_| |_|_| |_|_|
|_|_| | |_| | |_| | ._| |_|_| |_|_| |_| | |_| | |_. | |_|_| |_|_|
|_|_| |___| |___| | |_| | ._| |_|_| |___| |___| |_| | |_. | |_|_|
|_|_| | ._| |_|_| |___| | |_| | ._| |_. | |_|_| |___| |_| | |_. |
|_|_| | |_| |_|_| |_|_| |___| | |_| |_| | |_|_| |_|_| |___| |_| |
|_|_| |___| |_|_| |_|_| |_|_| |___| |___| |_|_| |_|_| |_|_| |___|
.___. .___. .___. .___. .___. .___. .___. .___. .___. .___. .___.
| ._| |_. | | |_| |_|_| |_|_| |_. | |_. | |_|_| |_. | |_|_| |_|_|
| |_| |_| | | |_| | |_| |_|_| | | | |_| | |_. | |_| | |_. | |_|_|
|___| |___| | |_| | |_| | |_| | | | | | | | | | |_| | |_| | |_. |
|_. | | ._| |___| | |_| | |_| | |_| | |_| | | | |_|_| |_| | |_| |
|_| | | |_| |_|_| |___| | |_| |___| | |_| | |_| |_|_| |_|_| |_| |
|___| |___| |_|_| |_|_| |___| |_|_| |___| |___| |_|_| |_|_| |_|_|
.___. .___. .___. .___. .___. .___. .___. .___. .___.
| ._| |_|_| |_|_| | ._| | ._| |_|_| |_|_| |_|_| |_| |
| |_| | ._| |_|_| | | | | |_| | ._| |_|_| |_| | |_| |
| |_| | |_| | ._| | | | | | | | | | |_| | |_| | |_| |
|_|_| | |_| | |_| |_| | |_| | | | | |_| | |_| | |___|
|_|_| |_|_| | |_| |___| |_| | |_| | |_| | |___| |_|_|
|_|_| |_|_| |_|_| |_|_| |___| |___| |___| |_|_| |_|_|  .

Links

Crossrefs

Cf. A247125, A278330, A278456, A278657, A278877, A286391.

A286347 Number of tilings of a 6 X n rectangle using pentominoes of shapes X, Y, T and monominoes.

Original entry on oeis.org

1, 1, 15, 188, 3763, 54368, 790504, 11771179, 176285795, 2627027061, 39172906410, 584774465361, 8725442013820, 130117410190768, 1940656499251689, 28950613608637003, 431889355175650435, 6442566792491842258, 96102610815876310611, 1433558139553752297236
Offset: 0

Views

Author

Alois P. Heinz, May 08 2017

Keywords

Links

Crossrefs

Cf. A247125, A278330, A278456, A278657, A278877, A278964, A286391, A321589.

A286391 Number of tilings of a 6 X n rectangle using pentominoes of shapes X, Y, Z, T and monominoes.

Original entry on oeis.org

1, 1, 15, 344, 6835, 119392, 2136080, 38329669, 688355077, 12374676591, 222673114420, 4005005917279, 72004800735036, 1294444641691072, 23273376392535695, 418470613768822683, 7524342117053834729, 135288703461639048592, 2432489164334993344213
Offset: 0

Views

Author

Alois P. Heinz, May 08 2017

Keywords

Examples

			a(2) = 15:
.___. .___. .___. .___. .___. .___. .___. .___. .___.
|_|_| | |_| |_| | | |_| |_|_| |_|_| |_|_| |_|_| | |_|
|_|_| | ._| |_. | | ._| | |_| |_|_| |_|_| | |_| | |_|
|_|_| | | | | | | | |_| | ._| | |_| | |_| | |_| | ._|
|_|_| |_| | | |_| |_|_| | |_| | ._| | |_| | ._| |_|_|
|_|_| |_. | | ._| |_|_| |_|_| | |_| | ._| |_|_| |_|_|
|_|_| |_|_| |_|_| |_|_| |_|_| |_|_| |_|_| |_|_| |_|_|
.___. .___. .___. .___. .___. .___.
|_| | |_|_| |_|_| |_|_| |_|_| |_| |
|_. | |_| | |_|_| |_|_| |_| | |_| |
|_| | |_. | |_| | |_| | |_| | |_. |
|_|_| |_| | |_. | |_| | |_. | |_|_|
|_|_| |_|_| |_| | |_. | |_|_| |_|_|
|_|_| |_|_| |_|_| |_|_| |_|_| |_|_|  .

Links

Crossrefs

Cf. A247125, A278330, A278456, A278657, A278877, A278964, A286347.

A278874 Number of tilings of a 2 X n rectangle using pentominoes of any shape and monominoes.

Original entry on oeis.org

1, 1, 1, 7, 25, 50, 155, 508, 1343, 3800, 11438, 32525, 92333, 268766, 774302, 2216976, 6392865, 18425916, 52958070, 152425812, 438973764, 1263109849, 3634965137, 10463959311, 30116734921, 86675829307, 249478723992, 718056248229, 2066658063664, 5948257601097
Offset: 0

Views

Author

Alois P. Heinz, Nov 29 2016

Keywords

Examples

			a(3) = 7:
._____. ._____. ._____. ._____. ._____. ._____. ._____.
|_|_|_| | |_| | | ._. | |   ._| |_.   | |   |_| |_|   |
|_|_|_| |_____| |_|_|_| |___|_| |_|___| |_____| |_____| .

Links

Crossrefs

Column k=2 of A278657.

Formula

G.f.: -(x^10 +x^8 -x^6 -6*x^5 -x^4 -5*x^3 +1) / (x^15 +x^13 -2*x^11 -11*x^10 -2*x^9 -10*x^8 +x^7 +9*x^6 +12*x^5 +8*x^4 +11*x^3 +x -1).

A278875 Number of tilings of a 3 X n rectangle using pentominoes of any shape and monominoes.

Original entry on oeis.org

1, 1, 7, 50, 311, 1954, 11914, 76003, 467221, 2943211, 18261840, 114360149, 712196192, 4449548684, 27749537725, 173227638835, 1080825788517, 6745415139188, 42092502492537, 262680587755837, 1639226363457986, 10229514197548963, 63836523795617745
Offset: 0

Views

Author

Alois P. Heinz, Nov 29 2016

Keywords

Examples

			a(2) = 7:
.___.  .___.  .___.  .___.  .___.  .___.  .___.
|_|_|  |   |  |   |  | |_|  |_| |  | ._|  |_. |
|_|_|  | ._|  |_. |  |   |  |   |  | |_|  |_| |
|_|_|  |_|_|  |_|_|  |___|  |___|  |___|  |___| .

Links

Crossrefs

Column k=3 of A278657.

Formula

G.f.: -(x^70 +9*x^66 -58*x^65 +2*x^64 +33*x^62 -442*x^61 +1350*x^60 +177*x^59 +x^58 -1800*x^57 +7590*x^56 -14372*x^55 -5301*x^54 -4274*x^53 +27565*x^52 -57859*x^51 +81976*x^50 +25142*x^49 +52927*x^48 -178142*x^47 +267283*x^46 -286938*x^45 +18226*x^44 -271940*x^43 +645655*x^42 -741357*x^41 +769892*x^40 -399635*x^39 +731247*x^38 -1389200*x^37 +1406759*x^36 -1517890*x^35 +1113060*x^34 -1250164*x^33 +1807581*x^32 -1790865*x^31 +1917897*x^30 -1559114*x^29 +1282018*x^28 -1412376*x^27 +1442106*x^26 -1364028*x^25 +1161216*x^24 -797934*x^23 +646269*x^22 -590362*x^21 +518438*x^20 -428452*x^19 +276264*x^18 -167608*x^17 +99537*x^16 -80445*x^15 +63323*x^14 -44329*x^13 +18699*x^12 -9385*x^11 -862*x^10 -2736*x^9 -120*x^8 -534*x^7 -609*x^6 -199*x^5 -37*x^4 -13*x^3 -16*x^2 +1)
/ (x^75 +10*x^71 -67*x^70 +21*x^67 -575*x^66 +1781*x^65 +497*x^64 -87*x^63 -1909*x^62 +11113*x^61 -20753*x^60 -13155*x^59 -6937*x^58 +36701*x^57 -96686*x^56 +136888*x^55 +78888*x^54 +85389*x^53 -331892*x^52 +555027*x^51 -523802*x^50 -79591*x^49 -653455*x^48 +1705482*x^47 -1806940*x^46 +1752446*x^45 -1137530*x^44 +2652875*x^43 -4859649*x^42 +4471399*x^41 -5140040*x^40 +4864686*x^39 -6183059*x^38 +9006019*x^37 -8490050*x^36 +9607056*x^35 -9441593*x^34 +8984384*x^33 -10781348*x^32 +10676975*x^31 -10736664*x^30 +10043865*x^29 -7885992*x^28 +8169411*x^27 -7730756*x^26 +6657930*x^25 -5913652*x^24 +4235643*x^23 -3306372*x^22 +2709439*x^21 -2366644*x^20 +1645722*x^19 -1148473*x^18 +713958*x^17 -384641*x^16 +318569*x^15 -200025*x^14 +134362*x^13 -53508*x^12 +39041*x^11 -2080*x^10 +6477*x^9 +903*x^8 +1940*x^7 +863*x^6 +394*x^5 +110*x^4 +34*x^3 +22*x^2 +x -1).

A278876 Number of tilings of a 4 X n rectangle using pentominoes of any shape and monominoes.

Original entry on oeis.org

1, 1, 25, 311, 4101, 56864, 728857, 9616962, 127040707, 1669454156, 22011707486, 289906000417, 3817261693883, 50281947199556, 662232832054368, 8721984517729672, 114875834939622917, 1512983475914685706, 19927016855891011234, 262452496836729312910
Offset: 0

Views

Author

Alois P. Heinz, Nov 29 2016

Keywords

Examples

			a(2) = 25:
.___. .___. .___. .___. .___. .___. .___. .___.
| ._| |_|_| |_. | |_|_| | |_| |_| | | ._| |_. |
| |_| | ._| |_| | |_. | | |_| |_| | | |_| |_| |
|___| | |_| |___| |_| | | |_| |_| | | |_| |_| |
|_|_| |___| |_|_| |___| |___| |___| |_|_| |_|_|
.___. .___. .___. .___. .___. .___. .___. .___.
|   | |_|_| |   | |_|_| | |_| |_|_| |_| | |_|_|
| ._| |   | |_. | |   | |   | | |_| |   | |_| |
|_|_| | ._| |_|_| |_. | |___| |   | |___| |   |
|_|_| |_|_| |_|_| |_|_| |_|_| |___| |_|_| |___|
.___. .___. .___. .___. .___. .___. .___. .___. .___.
| |_| |_| | | |_| |_| | | |_| |_| | |_| | | |_| |_|_|
| |_| |_| | | ._| |_. | | |_| |_| | | ._| |_. | |_|_|
| ._| |_. | | |_| |_| | |_. | | ._| | |_| |_| | |_|_|
|_|_| |_|_| |_|_| |_|_| |_|_| |_|_| |_|_| |_|_| |_|_| .

Links

Crossrefs

Column k=4 of A278657.

A278878 Number of tilings of a 7 X n rectangle using pentominoes of any shape and monominoes.

Original entry on oeis.org

1, 4, 508, 76003, 9616962, 1178422563, 147382229321, 18446349150367, 2306845502243104, 288660593026069762, 36110872742261362007, 4516714276347536995415, 564976767651757484324113, 70671912291059592748455901, 8840222401663200361818980813
Offset: 0

Views

Author

Alois P. Heinz, Nov 29 2016

Keywords

Examples

			a(1) = 4:
._.   ._.   ._.   ._.
|_|   | |   |_|   |_|
|_|   | |   | |   |_|
|_|   | |   | |   | |
|_|   | |   | |   | |
|_|   |_|   | |   | |
|_|   |_|   |_|   | |
|_|   |_|   |_|   |_|  .

Links

Crossrefs

Column k=7 of A278657.
Showing 1-10 of 10 results.

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