A233427 -id:A233427 - cp's OEIS frontend

A264765 Number of tilings of a 5 X n rectangle using n pentominoes of shapes Z, I, P.

1, 1, 3, 7, 17, 78, 195, 616, 1783, 5120, 16714, 48843, 150407, 453178, 1356478, 4174538, 12554221, 38233788, 115868736, 350343710, 1065875246, 3225913135, 9793613873, 29699991965, 90011535049, 273180669975, 828073217940, 2511974932751, 7618229843186

Offset: 0

Alois P. Heinz, Nov 23 2015

nonn

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

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

Cf. A174249, A233427, A234931, A247268, A247443, A249762, A264812.

A247117 Number of tilings of a 10 X n rectangle using 2n pentominoes of shape I.

Original entry on oeis.org

1, 1, 1, 1, 1, 8, 17, 28, 41, 56, 144, 317, 609, 1060, 1716, 3324, 6713, 13188, 24624, 43620, 80464, 153645, 296025, 562097, 1037921, 1920661, 3600832, 6820873, 12920804, 24211457, 45173688, 84493668, 158848825, 299451277, 562923960, 1055117520, 1976475968

Offset: 0

Alois P. Heinz, Nov 19 2014

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

Cf. A174249, A233427, A003520 (5 X n), A247218 (15 X n).

Column k=5 of A250662.

gf:= -(x^10+x^8-x^6-2*x^5-x^4-x^3+1) *(x-1)^4 *(x^4+x^3+x^2+x+1)^4 / (x^35 +x^33 -2*x^31 -7*x^30 -2*x^29 -6*x^28 +x^27 +9*x^26 +22*x^25 +8*x^24 +15*x^23 -4*x^22 -15*x^21 -39*x^20 -12*x^19 -20*x^18 +6*x^17 +10*x^16 +45*x^15 +8*x^14 +19*x^13 -4*x^12 -4*x^11 -33*x^10 -6*x^9 -10*x^8 +x^7 -3*x^6 +12*x^5 +x^3 +x-1):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..50);

G.f.: see Maple program.

A247126 Number of tilings of a 5 X n rectangle using n pentominoes of shapes F, U, X, N.

Original entry on oeis.org

1, 0, 0, 1, 2, 0, 1, 4, 4, 1, 14, 12, 17, 32, 64, 81, 138, 272, 489, 764, 1548, 2809, 5062, 9420, 17721, 32712, 60992, 114105, 213890, 398784, 747745, 1401476, 2624004, 4916369, 9218118, 17274340, 32378521, 60694768, 113785984, 213293721, 399856922, 749628208

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, A234931, A077909, A247127.

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

G.f.: see Maple program.

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.

A247103 Number of tilings of a 5 X n rectangle using n pentominoes of shapes I, N, P, U, T.

Original entry on oeis.org

1, 1, 3, 11, 33, 166, 589, 2216, 8935, 33984, 137056, 539085, 2100341, 8324716, 32607928, 128652672, 507032667, 1992083368, 7853132654, 30894420646, 121642017784, 479048967517, 1885229164497, 7423732617043, 29223734864421, 115048209160729, 452968829090506

Offset: 0

Alois P. Heinz, Nov 20 2014

nonn

			a(3) = 11:
._____.     ._____.     ._____.     ._____.
| | | |     |   | |     | |   |     |_.   |
| | | |     | ._| |     | | ._|     | |___|
| | | |     |_| ._|     | |_| |     | .___|
| | | |     | |_| |     | |   |     |_|   |
|_|_|_| (1) |_____| (4) |_|___| (4) |_____| (2) .

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

Cf. A174249, A233427, A247196.

A247196 Number of tilings of a 5 X n rectangle using n pentominoes of shapes T, I, P.

Original entry on oeis.org

1, 1, 3, 7, 17, 78, 199, 638, 1865, 5332, 17250, 50877, 156787, 475348, 1423432, 4367178, 13182353, 40122074, 121916294, 369052634, 1122475534, 3403574961, 10335095973, 31385295907, 95218937465, 289154182737, 877572289140, 2663965244527, 8087517963748

Offset: 0

Alois P. Heinz, Nov 24 2014

nonn

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

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

Cf. A174249, A233427, A246764, A247103.

A247218 Number of tilings of a 15 X n rectangle using 3n pentominoes of shape I.

Original entry on oeis.org

1, 1, 1, 1, 1, 34, 95, 190, 325, 506, 3324, 10353, 25607, 55346, 108756, 389216, 1208901, 3281686, 8006108, 17950204, 51430928, 150609259, 419540401, 1090827453, 2651884943, 7077981621, 19691707908, 54499735145, 145671654672, 371632691473, 976543067070

Offset: 0

Alois P. Heinz, Nov 26 2014

nonn

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

Cf. A174249, A233427, A003520, A247117.

Column k=5 of A251072.

A251617 Number of tilings of a 5 X n rectangle using n pentominoes of shapes F, I, L, U.

Original entry on oeis.org

1, 1, 3, 5, 19, 74, 219, 628, 1749, 5486, 17448, 53383, 160169, 479908, 1468366, 4512092, 13782535, 41855766, 127112554, 387469920, 1182800866, 3606789463, 10983721059, 33445214911, 101911804705, 310658892951, 946813182854, 2884825285301, 8789233684468

Offset: 0

Alois P. Heinz, Dec 05 2014

nonn

			a(4) = 19 = 13 + 4 + 2 = A249762(4) + 4 + 2:
._______.     ._______.
|_____. |     | ._____|
| | ._|_|     |_| ._. |
| | |_. |     | |_| |_|
| |___| |     |_____| |
|___|___| (4) |_______| (2) .

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

Cf. A174249, A233427, A247443, A247680, A249762, A251737.

A251737 Number of tilings of a 5 X n rectangle using n pentominoes of shapes L, U, I.

Original entry on oeis.org

1, 1, 3, 5, 17, 66, 181, 508, 1283, 3664, 10812, 31171, 88565, 245524, 692416, 1968532, 5609977, 15928174, 44982196, 127190716, 360208608, 1021611491, 2896270245, 8202605953, 23226285083, 65780006703, 186369631872, 528047092459, 1495905404102, 4237308534243

Offset: 0

Alois P. Heinz, Dec 07 2014

nonn

			a(4) = 17:
._______.     ._______.     ._______.     ._______.
|_____. |     | ._____|     | | | | |     | | ._| |
| | ._|_|     |_| ._. |     | | | | |     | | | | |
| | |_. |     | |_| |_|     | | | | |     | | | | |
| |___| |     |_____| |     | | | | |     | |_| | |
|___|___| (4) |_______| (2) |_|_|_|_| (1) |_|___|_| (2)
._______.     ._______.     ._______.
| ._| | |     | ._| ._|     | ._|_. |
| | | | |     | | | | |     | | | | |
| | | | |     | | | | |     | | | | |
|_| | | |     |_| |_| |     |_| | |_|
|___|_|_| (4) |___|___| (2) |___|___| (2) .

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

Cf. A174249, A233427, A247443, A247680, A249762, A251617.

A257866 Number of tilings of a 5 X n rectangle using n pentominoes of shapes W, I, L.

Original entry on oeis.org

1, 1, 3, 5, 19, 74, 209, 572, 1479, 4304, 13002, 38315, 109651, 308982, 884120, 2560952, 7428183, 21413028, 61433280, 176415916, 507985116, 1464725431, 4220293147, 12145885239, 34945690653, 100586823613, 289649303130, 834087280681, 2401368817168, 6912685066843

Offset: 0

Alois P. Heinz, May 11 2015

nonn

			a(3) = 5:
._____. ._____. ._____. ._____. ._____.
| | | | | |_. | | ._| | | | ._| |_. | |
| | | | | | | | | | | | | | | | | | | |
| | | | | | | | | | | | | | | | | | | |
| | | | | | |_| |_| | | | |_| | | |_| |
|_|_|_| |_|___| |___|_| |_|___| |___|_|.
a(4) = 19:
._______. ._______.
|_. |_. | | | ._| |
| |_. | | | | | | |
|_. |_| | | | | | |
| |___|_| | |_| | |
|_______| |_|___|_| ... .

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, G.f. and the recurrence (of order 315)
Wikipedia, Pentomino

Cf. A174249, A233427, A234312, A234931, A247443, A247680, A249762.

a(n) ~ c * d^n, where d = 2.878962978866730659679600165158895088546680936475540731494833253735549346144..., c = 0.33249894796240209167801000207088312509480543003269025485052861968247997... (1/d is the root of the denominator, see g.f.). - Vaclav Kotesovec, May 19 2015

cp's OEIS Frontend

A264765 Number of tilings of a 5 X n rectangle using n pentominoes of shapes Z, I, P.

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A247117 Number of tilings of a 10 X n rectangle using 2n pentominoes of shape I.

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Maple

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A247126 Number of tilings of a 5 X n rectangle using n pentominoes of shapes F, U, X, N.

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Maple

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

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A247103 Number of tilings of a 5 X n rectangle using n pentominoes of shapes I, N, P, U, T.

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A247196 Number of tilings of a 5 X n rectangle using n pentominoes of shapes T, I, P.

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A247218 Number of tilings of a 15 X n rectangle using 3n pentominoes of shape I.

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A251617 Number of tilings of a 5 X n rectangle using n pentominoes of shapes F, I, L, U.

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A251737 Number of tilings of a 5 X n rectangle using n pentominoes of shapes L, U, I.

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A257866 Number of tilings of a 5 X n rectangle using n pentominoes of shapes W, I, L.

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