A174249 -id:A174249 - cp's OEIS frontend

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

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

A352421 Number of tilings of a 5 X n rectangle using n pentominoes of shapes U, Y, Z.

Original entry on oeis.org

1, 0, 0, 0, 0, 4, 6, 8, 6, 8, 54, 112, 182, 232, 404, 930, 2054, 3880, 6304, 10696, 20696, 42396, 81554, 146240, 259534, 480084, 924860, 1768856, 3284468, 5992798, 11044774, 20756310, 39209398, 73369392, 135855648, 251495794, 468915328, 878762056, 1644145874

Offset: 0

Alois P. Heinz, Apr 25 2022

			a(7) = 8:
  ._____________.     ._____________.     ._____________.
  |_. .___| | ._|     |_. .___| ._| |     | |_. .___| ._|
  | |_| .___| | |     | |_|_. | |_. |     | ._|_| ._| | |
  | ._|_| |___| |     | |_. | |___| |     | | ._| |___| |
  | | .___| |_. | (4) | ._| |___| |_| (2) |_| |___| |_. | (2)
  |_|_|_______|_|     |_|___|_______|     |___|_______|_|      .
.

Alois P. Heinz, Table of n, a(n) for n = 0..3699
Wikipedia, Pentomino
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 3, 8, 8, 12, 4, 2, 11, 10, 5, -1, -29, -90, -65, -20, 39, -13, -73, -71, -110, -128, -67, 44, 82, 0, -53, 67, 70, 61, -79, -96, -3, 79, 84, 64, 30, -42, -29, -44, 24, 0, 0, 2).

Cf. A174249, A343529, A349187.

G.f.: (16*x^42 -16*x^41 +3*x^40 -20*x^39 -28*x^38 +28*x^37 -2*x^36 +31*x^35 +15*x^34 -38*x^33 -29*x^32 +9*x^31 +20*x^30 +69*x^29 +3*x^28 -10*x^27 +10*x^26 +2*x^25 +31*x^24 -10*x^23 -20*x^22 -41*x^21 -37*x^20 -25*x^19 +7*x^18 +8*x^17 -3*x^16 -28*x^15 -31*x^14 -9*x^13 +x^12 +2*x^11 +7*x^10 +6*x^9 -2*x^8 +4*x^7 +2*x^6 +4*x^5 +3*x^4 -1) / (2*x^45 +24*x^42 -44*x^41 -29*x^40 -42*x^39 +30*x^38 +64*x^37 +84*x^36 +79*x^35 -3*x^34 -96*x^33 -79*x^32 +61*x^31 +70*x^30 +67*x^29 -53*x^28 +82*x^26 +44*x^25 -67*x^24 -128*x^23 -110*x^22 -71*x^21 -73*x^20 -13*x^19 +39*x^18 -20*x^17 -65*x^16 -90*x^15 -29*x^14 -x^13 +5*x^12 +10*x^11 +11*x^10 +2*x^9 +4*x^8 +12*x^7 +8*x^6 +8*x^5 +3*x^4 -1).

A361250 Number of tilings of a 5 X n rectangle using n pentominoes of shapes T, N, X.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 2, 2, 8, 0, 18, 6, 16, 6, 48, 22, 74, 48, 182, 74, 306, 204, 544, 342, 1114, 826, 2038, 1546, 4144, 3126, 7452, 6470, 14538, 12542, 27824, 25994, 53398, 50244, 103288, 101306, 195756, 200120, 380310, 395802

Offset: 0

Alois P. Heinz, Apr 20 2023

			a(10) = 2:
   .___________________.
   |___. |_. ._| .___| |
   |_. |___| |___|___  |
   | |_____|_| |___. |_|
   | .___| ._| |_. |___|
   |_|_____|_____|_____|  ... and its mirror.
.
a(14) = 2:
   .___________________________.
   |___. |_. ._| |_. ._| .___| |
   |_. |___| |_. ._| |___|___  |
   | |_____|_| |_| |_| |___. |_|
   | .___| ._| |_. ._| |_. |___|
   |_|_____|_____|_|_____|_____|  ... and its mirror.
.
a(16) = 2:
   ._______________________________.
   |___. |_. ._| .___|_. ._| .___| |
   |_. |___| |___| .___| |___|___  |
   | |_____|_| |___| ._|_| |___. |_|
   | .___| ._| |_____| ._| |_. |___|
   |_|_____|_____|_____|_____|_____|  ... and its mirror.
.
a(17) = 2:
   ._________________________________.
   |___. |_. ._| |___. |_. ._| .___| |
   |_. |___| |_. ._| |___| |___|___  |
   | |_____|_| |_|_. ._| |_| |___. |_|
   | .___| ._| |_. |_|_. ._| |_. |___|
   |_|_____|_____|_____|_|_____|_____|  ... and its mirror.
.

Alois P. Heinz, Table of n, a(n) for n = 0..5000
Wikipedia, Pentomino
Index entries for linear recurrences with constant coefficients, signature (0,0,1,1,0,1,0,3,-1,6,-2,0,-6,-1).

Cf. A174249, A343529, A349187, A352421, A358933.

gf:= (x^14+4*x^13+2*x^11-4*x^10+x^9-3*x^8-x^6-x^4-x^3+1)/
     (x^14+6*x^13+2*x^11-6*x^10+x^9-3*x^8-x^6-x^4-x^3+1):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..66);

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.

cp's OEIS Frontend

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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A352421 Number of tilings of a 5 X n rectangle using n pentominoes of shapes U, Y, Z.

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A361250 Number of tilings of a 5 X n rectangle using n pentominoes of shapes T, N, X.

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Maple

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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Maple

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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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