A233427 -id:A233427 - cp's OEIS frontend

A233428 Number of tilings of a 3 X 5n rectangle with 3n pentominoes of any shape.

1, 56, 7670, 1082911, 153054306, 21632866165, 3057616805283, 432167457281031, 61083099363540480, 8633563135788393662, 1220278820098831948033, 172475764103555415010594, 24377944378888377125376221, 3445609736701113995818305965, 487006872816193035818432289071

Offset: 0

Alois P. Heinz, Dec 09 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..200
Wikipedia, Pentomino

Quintisection of column k=3 of A233427.

From Vaclav Kotesovec, Mar 05 2016: (Start)
a(n) ~ c * d^n, where d = 141.34127484863151940788399760068559708960763498273966116022774034..., c = 0.3835032349236650628846889495224683008372791393401511291113817887...
a(n) = 172*a(n-1) - 4716*a(n-2) + 56595*a(n-3) - 364164*a(n-4) + 1353076*a(n-5) - 3014276*a(n-6) + 4180766*a(n-7) - 3711813*a(n-8) + 2129818*a(n-9) - 781787*a(n-10) + 178168*a(n-11) - 24000*a(n-12) + 1780*a(n-13) - 67*a(n-14) + a(n-15).
G.f.: (-1 + 116*x - 2754*x^2 + 28828*x^3 - 160178*x^4 + 509733*x^5 - 963854*x^6 + 1114401*x^7 - 801386*x^8 + 358357*x^9 - 97595*x^10 + 15483*x^11 - 1335*x^12 + 58*x^13 - x^14)/( - 1 + 172*x - 4716*x^2 + 56595*x^3 - 364164*x^4 + 1353076*x^5 - 3014276*x^6 + 4180766*x^7 - 3711813*x^8 + 2129818*x^9 - 781787*x^10 + 178168*x^11 - 24000*x^12 + 1780*x^13 - 67*x^14 + x^15).
(End)

A233429 Number of tilings of a 4 X 5n rectangle with 4n pentominoes of any shape.

Original entry on oeis.org

1, 501, 890989, 1666772813, 3125064284901, 5859766228545967, 10987595369808657169, 20602741350971679371521, 38632014757792375332001835, 72438542874037975287077565935, 135828859204452470219311220073385, 254691470323770059993623356733063003

Offset: 0

Alois P. Heinz, Dec 09 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..100
Wikipedia, Pentomino

Quintisection of column k=4 of A233427.

a(n) ~ c * d^n, where d = 1875.09099182186211660253860201132854810972459605757360689785461004..., c = 0.25279505771792803447854701369537504374767224177480509542292622525... . - Vaclav Kotesovec, Mar 05 2016

A246764 Number of tilings of a 5 X n rectangle using n pentominoes of shapes P, I, N, T.

Original entry on oeis.org

1, 1, 3, 7, 17, 78, 247, 916, 3301, 11272, 41854, 150485, 538585, 1954912, 6978464, 25170446, 90851829, 326048198, 1176355862, 4230352602, 15222263126, 54855015353, 197302183497, 710556528403, 2557837610375, 9205575728179, 33148388282116, 119307072980025

Offset: 0

Alois P. Heinz, Nov 28 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Pentomino

Cf. A174249, A233427, A247103, A247196.

A246902 Number of tilings of a 5 X n rectangle using n pentominoes of distinct shapes.

Original entry on oeis.org

1, 1, 0, 28, 200, 856, 2164, 5584, 13632, 23608, 27804, 16412, 4040

Offset: 0

Alois P. Heinz, Nov 16 2014

			a(3) = 28: 4 orientations of each of the following 7 patterns:
  ._____. ._____. ._____. ._____. ._____. ._____. ._____.
  |___. | | |   | |_. ._| | .___| |   | | |_.   | | ._| |
  | | | | | | ._| | | | | | |   | | ._| | | |___| | |_. |
  | | |_| | |_| | | |_| | |_| ._| |_| ._| |_. ._| |___| |
  | |_. | |___| | | |_. | | |_| | | |_| | | |_| | |   |_|
  |___|_| |_____| |___|_| |_____| |_____| |_____| |_____| .

Wikipedia, Pentomino

Cf. A174249, A233427.

A247121 Number of tilings of a 5 X 2n rectangle using 2n pentominoes of shapes P, U.

Original entry on oeis.org

1, 2, 12, 56, 248, 1184, 5472, 25376, 118208, 548864, 2550912, 11856896, 55098368, 256070144, 1190065152, 5530658816, 25703241728, 119453057024, 555145224192, 2579979739136, 11990182412288, 55723107221504, 258967268524032, 1203523043065856, 5593246378754048

Offset: 0

Alois P. Heinz, Nov 19 2014

nonn

			a(2) = 12:
._______.      ._______.      ._______.      ._______.
|   |   |      |   ._| |      | ._|   |      | ._|   |
| ._| ._|      |___|   |      | |_____|      | |_____|
|_| |_| |      |   |___|      |___|   |      |___|_. |
|   |   |      | ._|   |      |   |_. |      |   ._| |
|___|___| (*4) |_|_____| (*2) |_____|_| (*4) |___|___| (*2) .

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Pentomino
Index entries for linear recurrences with constant coefficients, signature (2,8,20).

Cf. A174249, A233427, A247076.

a:= n-> ceil((<<0|1|0>, <0|0|1>, <20|8|2>>^(n-1). <<2, 12, 56>>)[1, 1]):
seq(a(n), n=0..30);

G.f.: (4*x^3-1)/(20*x^3+8*x^2+2*x-1).

A247127 Number of tilings of a 5 X n rectangle using n pentominoes of shapes V, U, X, N.

Original entry on oeis.org

1, 0, 0, 1, 4, 0, 9, 8, 24, 17, 78, 64, 227, 212, 664, 699, 2004, 2220, 6033, 7196, 18112, 22859, 54882, 72560, 166251, 229284, 505632, 721421, 1540532, 2264668, 4702135, 7092742, 14376450, 22165709, 44024116, 69154334, 134973515, 215459398, 414268932

Offset: 0

Alois P. Heinz, Nov 19 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Pentomino

Cf. A174249, A233427, A247126.

gf:= -(4*x^18 +4*x^17 -8*x^16 -3*x^15 -9*x^14 +2*x^13 -3*x^12 +5*x^11 -7*x^10 +x^9 -7*x^8 -x^6 -2*x^5 -x^3+1) / (32*x^26 +32*x^25 -32*x^24 +8*x^23 -120*x^22 +12*x^21 -124*x^20 +36*x^19 -123*x^18 +35*x^17 -106*x^16 +20*x^15 -62*x^14 -23*x^13 -22*x^12 -36*x^11 +5*x^10 -18*x^9 +13*x^8 -4*x^7 +8*x^6 +2*x^5 +4*x^4 +2*x^3-1):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..50);

G.f.: see Maple program.

A247735 Total number of F shapes in all tilings of a 5 X n rectangle with pentominoes of any shape.

Original entry on oeis.org

0, 0, 0, 4, 68, 744, 7088, 69176, 658776, 5995348, 52831448, 458439732, 3944077760, 33606901296, 283630894720, 2375471834288, 19779055344468, 163866917256420, 1351464466008112, 11100931884955344, 90860898121739864, 741378713523090200, 6032337718479599100

Offset: 0

Alois P. Heinz, Sep 23 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..500
Wikipedia, Pentomino

Cf. A174249, A233427, A247702.

a(n) = Sum_{k>0} k * A247702(n,k).

A247736 Total number of I shapes in all tilings of a 5 X n rectangle with pentominoes of any shape.

Original entry on oeis.org

0, 1, 2, 11, 122, 2278, 17502, 140037, 1250890, 11184764, 96901618, 810143038, 6787465080, 56871594099, 473417649550, 3913478050583, 32192184170708, 264014564271675, 2158561440821352, 17592866753339355, 142983916007396542, 1159269824603687631, 9378694043134778568

Offset: 0

Alois P. Heinz, Sep 23 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..500
Wikipedia, Pentomino

Cf. A174249, A233427, A247703.

a(n) = Sum_{k>0} k * A247703(n,k).

A247737 Total number of L shapes in all tilings of a 5 X n rectangle with pentominoes of any shape.

Original entry on oeis.org

0, 0, 4, 24, 436, 3888, 29880, 264460, 2353136, 20400352, 171215320, 1430108512, 11962161392, 99523393628, 822501750184, 6763177931936, 55433124540308, 453013557727460, 3690931980743340, 29988435689294280, 243064471330287696, 1965884816120576180

Offset: 0

Alois P. Heinz, Sep 23 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..500
Wikipedia, Pentomino

Cf. A174249, A233427, A247704.

a(n) = Sum_{k>0} k * A247704(n,k).

A247738 Total number of N shapes in all tilings of a 5 X n rectangle with pentominoes of any shape.

Original entry on oeis.org

0, 0, 0, 8, 88, 1024, 10388, 99840, 935900, 8415368, 74295280, 645535244, 5545784444, 47152918764, 397457338488, 3327197881716, 27688757985292, 229256300558340, 1889667799688076, 15514699106805468, 126939424800685264, 1035396864840877088, 8421920979218715332

Offset: 0

Alois P. Heinz, Sep 23 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..500
Wikipedia, Pentomino

Cf. A174249, A233427, A247705.

a(n) = Sum_{k>0} k * A247705(n,k).

cp's OEIS Frontend

A233428 Number of tilings of a 3 X 5n rectangle with 3n pentominoes of any shape.

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A233429 Number of tilings of a 4 X 5n rectangle with 4n pentominoes of any shape.

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A246764 Number of tilings of a 5 X n rectangle using n pentominoes of shapes P, I, N, T.

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A246902 Number of tilings of a 5 X n rectangle using n pentominoes of distinct shapes.

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A247121 Number of tilings of a 5 X 2n rectangle using 2n pentominoes of shapes P, U.

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A247127 Number of tilings of a 5 X n rectangle using n pentominoes of shapes V, U, X, N.

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A247735 Total number of F shapes in all tilings of a 5 X n rectangle with pentominoes of any shape.

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A247736 Total number of I shapes in all tilings of a 5 X n rectangle with pentominoes of any shape.

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A247737 Total number of L shapes in all tilings of a 5 X n rectangle with pentominoes of any shape.

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A247738 Total number of N shapes in all tilings of a 5 X n rectangle with pentominoes of any shape.

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