A247706 -id:A247706 - cp's OEIS frontend

A174249 Number of tilings of a 5 X n rectangle with n pentominoes of any shape.

1, 1, 5, 56, 501, 4006, 27950, 214689, 1696781, 13205354, 101698212, 782267786, 6048166230, 46799177380, 361683136647, 2793722300087, 21583392631817, 166790059833039, 1288885349447958, 9959188643348952, 76953117224941654, 594617039453764617, 4594660583890506956

Offset: 0

Bob Harris (me13013(AT)gmail.com), Mar 13 2010

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Jessica Gonzalez, Illustration for a(2) = 5
R. S. Harris, Counting Nonomino Tilings and Other Things of that Ilk, G4G9 Gift Exchange book, 2010.
R. S. Harris, Counting Polyomino Tilings
Vaclav Kotesovec, G.f. and the recurrence (of order 324)
Wikipedia, Pentomino
Index entries for linear recurrences with constant coefficients, order 324.

Cf. A134438, A174248, A174250, A174251, A174252, A174253, A246902, A278456.
Column k=5 of A233427.
Row sums of A247702, A247703, A247704, A247705, A247706, A247707, A247708, A247709, A247710, A247711, A247712, A247713.

a(n) ~ c * d^n, where d =
7.727036840800092392128639105511391434436212757335030092041375597587338371937..., c =
0.13364973920881772493778581621701653927538155984099992758656160782495174... (1/d is the root of the denominator, see g.f.). - Vaclav Kotesovec, May 19 2015

a(0) prepended, a(11)-a(22) from Alois P. Heinz, Dec 05 2013

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

Alois P. Heinz, Dec 09 2013

			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, ...
  ...

Liang Kai, Antidiagonals n = 0..26, flattened (Antidiagonals n = 0..17 from Alois P. Heinz)
R. S. Harris, Counting Polyomino Tilings
Liang Kai, Solving tiling enumeration problems by tensor network contractions, arXiv:2503.17698 [math.CO], 2025.
Wikipedia, Pentomino

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).

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

A247076 Number of tilings of a 5 X 2n rectangle using 2n pentominoes of shape P.

Original entry on oeis.org

1, 2, 6, 20, 62, 194, 612, 1922, 6038, 18980, 59646, 187442, 589076, 1851266, 5817894, 18283700, 57459518, 180575906, 567489348, 1783428098, 5604714422, 17613731780, 55354032894, 173959101458, 546694927604, 1718078222594, 5399341807686, 16968314698580

Offset: 0

Alois P. Heinz, Nov 17 2014

nonn

			a(2) = 6:
._______. ._______. ._______. ._______. ._______. ._______.
|   |   | |   |   | |   |   | |   |   | |   ._| | | |_.   |
| ._| ._| |_. |_. | | ._|_. | |_. | ._| |___|   | |   |___|
|_| |_| | | |_| |_| |_| | |_| | |_|_| | |   |___| |___|   |
|   |   | |   |   | |   |   | |   |   | | ._|   | |   |_. |
|___|___| |___|___| |___|___| |___|___| |_|_____| |_____|_| .

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Pentomino
Index entries for linear recurrences with constant coefficients, signature (2,2,5).

Even bisection of main diagonal of A247706.

Cf. A174249, A233427, A247121.

a:= proc(n) option remember; `if`(n<4, [1, 2, 6, 20][n+1],
       2*a(n-1) +2*a(n-2) +5*a(n-3))
    end:
seq(a(n), n=0..40);

Join[{1}, LinearRecurrence[{2, 2, 5}, {2, 6, 20}, 40]] (* Jean-François Alcover, May 29 2018 *)

G.f.: (x-1)*(x^2+x+1)/(5*x^3+2*x^2+2*x-1).

a(n) = 2*a(n-1)+2*a(n-2)+5*a(n-3) for n>3, a(0)=1; a(1)=2, a(2)=6, a(3)=20.

A247739 Total number of P shapes in all tilings of a 5 X n rectangle with pentominoes of any shape.

Original entry on oeis.org

0, 0, 4, 60, 556, 5328, 45316, 405636, 3583136, 30801264, 261702184, 2204389124, 18485952896, 154072120428, 1276431617512, 10525735832272, 86463238965520, 707822586795020, 5776135319900600, 47001025464956044, 381480686633266156, 3089168335328298424, 24963223479997963992

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, A247706.

a(n) = Sum_{k>0} k * A247706(n,k).

A247770 Number of tilings of a 5 X n rectangle using n pentominoes of any but the P shape.

Original entry on oeis.org

1, 1, 3, 16, 135, 944, 4814, 26435, 158761, 978044, 5847896, 34112584, 200686378, 1192024482, 7093151801, 42038865123, 248625856115, 1472077262971, 8724918986240, 51715149361624, 306384777098954, 1814848220902299, 10751681585808618, 63702848673263049

Offset: 0

Alois P. Heinz, Sep 23 2014

nonn

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

Column k=0 of A247706.

Cf. A174249, A233427.

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A174249 Number of tilings of a 5 X n rectangle with n pentominoes of any shape.

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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.

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A247076 Number of tilings of a 5 X 2n rectangle using 2n pentominoes of shape P.

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A247739 Total number of P shapes in all tilings of a 5 X n rectangle with pentominoes of any shape.

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A247770 Number of tilings of a 5 X n rectangle using n pentominoes of any but the P shape.

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