A174249 -id:A174249 - cp's OEIS frontend

A247712 Number T(n,k) of tilings of a 5 X n rectangle with pentominoes of any shape and exactly k pentominoes of shape Y; triangle T(n,k), n>=0, read by rows.

1, 1, 5, 44, 12, 321, 136, 44, 2404, 1160, 404, 24, 14, 14692, 9380, 3388, 392, 90, 8, 98831, 78492, 30834, 5724, 748, 60, 684729, 631020, 292250, 74016, 13280, 1428, 58, 4642752, 4944856, 2628566, 788284, 171368, 25648, 3648, 228, 4

Offset: 0

Alois P. Heinz, Sep 23 2014

Sum_{k>0} k * T(n,k) = A247745(n).

T(10*n,10*n) = 10^n = A011557(n).

			T(3,1) = 12:
._____.        ._____.        ._____.
| |_. |        |_.   |        | |_. |
| ._| |        | |___|        | ._| |
| | | |        | ._| |        | |___|
|_| |_|        | |   |        |_|   |
|_____| (*4)   |_|___| (*4)   |_____| (*4)  .
T(10,10) = 10:
.___________________.
|_. .___| |___. ._| |
| |_| |_______|_|_. |
| |_______|___. ._| |
| ._| |___. ._|_| |_|
|_|_______|_|_______| ... .
Triangle T(n,k) begins:
00 :      1;
01 :      1;
02 :      5;
03 :     44,     12;
04 :    321,    136,     44;
05 :   2404,   1160,    404,    24,    14;
06 :  14692,   9380,   3388,   392,    90,    8;
07 :  98831,  78492,  30834,  5724,   748,   60;
08 : 684729, 631020, 292250, 74016, 13280, 1428, 58;

Alois P. Heinz, Rows n = 0..145, flattened
Wikipedia, Pentomino

Row sums give A174249 or A233427(n,5).
Column k=0 gives A247776.
Cf. A011557, A247745.

A247713 Number T(n,k) of tilings of a 5 X n rectangle with pentominoes of any shape and exactly k pentominoes of shape Z; triangle T(n,k), n>=0, read by rows.

Original entry on oeis.org

1, 1, 5, 52, 4, 451, 48, 2, 3498, 484, 24, 23502, 4136, 300, 12, 173611, 37674, 3262, 142, 1323447, 335388, 35938, 1964, 44, 9920654, 2892492, 365458, 25752, 986, 12, 73573634, 24266128, 3544842, 298200, 15002, 400, 6, 545170514, 200531918, 33123244, 3236018, 198380, 7546, 164, 2

Offset: 0

Alois P. Heinz, Sep 23 2014

Sum_{k>0} k * T(n,k) = A247746(n).

			T(3,1) = 4:
._____.        ._____.
|___. |        |   ._|
|_. | |        |___| |
| | |_|        | .___|
| |___|        |_|   |
|_____| (*2)   |_____| (*2)  .
Triangle T(n,k) begins:
00 :        1;
01 :        1;
02 :        5;
03 :       52,        4;
04 :      451,       48,       2;
05 :     3498,      484,      24;
06 :    23502,     4136,     300,     12;
07 :   173611,    37674,    3262,    142;
08 :  1323447,   335388,   35938,   1964,    44;
09 :  9920654,  2892492,  365458,  25752,   986,  12;
10 : 73573634, 24266128, 3544842, 298200, 15002, 400, 6;

Alois P. Heinz, Rows n = 0..170, flattened
Wikipedia, Pentomino

Row sums give A174249 or A233427(n,5).
Column k=0 gives A247777.
Cf. A247746.

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

Original entry on oeis.org

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.

A343529 Number of tilings of a 5 X n rectangle using n pentominoes of shapes P, X, Y.

Original entry on oeis.org

1, 0, 2, 4, 18, 36, 138, 334, 1066, 3096, 9490, 26826, 80468, 235718, 699056, 2055466, 6074498, 17857906, 52725190, 155445504, 458505084, 1351257730, 3984941402, 11748306100, 34643781158, 102144907886, 301179533022, 887996181502, 2618324249106, 7720149428450

Offset: 0

Alois P. Heinz, Apr 18 2021

			a(2) = 2,        a(3) = 4:     a(5) = 36:
  .___.  .___.     ._____.       ._________.     ._________.
  |   |  |   |     |_.   |       |   |_.   |     |_. ._._| |
  | ._|  |_. |     | |___|       | ._| |___|     | |_| |_. |
  |_| |  | |_|     | ._| |       |_|_. ._| |     | |_. ._| |
  |   |  |   |     | |   | (4)   |   |_|   | (2) | ._|_| |_| (2)  ...
  |___|  |___|     |_|___|       |_____|___|     |_|_______|          .
.
a(4) = 18:
  .___.___.     .___.___.     ._._____.     ._______.
  |   |   |     |   |   |     | |_.   |     |___. ._|
  | ._|_. |     | ._| ._|     |   |___|     |   |_| |
  |_| | |_|     |_| |_| |     |___|   |     | ._|   |
  |   |   | (2) |   |   | (2) |   |_. | (2) |_| |___| (2)
  |___|___|     |___|___|     |_____|_|     |_______|
.
  ._______.     ._._____.     ._______.
  | |   ._|     | |_.   |     |___. ._|
  | |___| |     | ._|___|     |   |_| |
  | ._|_. |     | |_.   |     | ._|_. |
  |_|   | | (2) |_| |___| (4) |_|   | | (4)
  |_____|_|     |_______|     |_____|_|     .

Alois P. Heinz, Table of n, a(n) for n = 0..2131
Wikipedia, Pentomino
Index entries for linear recurrences with constant coefficients, signature (0, 4, 7, 16, 21, 19, -61, -198, -153, 399, 172, -511, -223, -1843, -1515, -702, 1445, 880, -6060, -13769, -3634, -761, -2886, 6196, -11350, -1559, -24546, 5474, 54995, 15927, -30841, 9893, 7921, 68397, -41768, -48989, 36892, 61021, 7095, -47120, -56180, 24792, 34252, 8426, -13154, -11260, 4136, 5668, -2228, -848, -1024, 224, 224, 144).

Cf. A174249, A247268, A264812, A278330.

G.f.: (16*x^54 +32*x^53 -128*x^51 -80*x^50 +380*x^49 +540*x^48 +456*x^47 -1316*x^46 -28*x^45 +976*x^44 +6016*x^43 +3356*x^42 -1680*x^41 -5992*x^40 -919*x^39 -825*x^38 +5838*x^37 -12209*x^36 -14876*x^35 -17029*x^34 -15243*x^33 -13879*x^32 -8029*x^31 -17115*x^30 -3713*x^29 -6022*x^28 -110*x^27 +1321*x^26 -832*x^25 -212*x^24 +4478*x^23 -575*x^22 -808*x^21 -3929*x^20 -574*x^19 +314*x^18 -1001*x^17 -1354*x^16 -805*x^15 -493*x^14 -299*x^13 -229*x^12 -78*x^11 +177*x^10 -39*x^9 -50*x^8 -19*x^7 +13*x^6 +15*x^5 +6*x^4 +3*x^3 +2*x^2 -1) /

(144*x^54 +224*x^53 +224*x^52 -1024*x^51 -848*x^50 -2228*x^49 +5668*x^48 +4136*x^47 -11260*x^46 -13154*x^45 +8426*x^44 +34252*x^43 +24792*x^42 -56180*x^41 -47120*x^40 +7095*x^39 +61021*x^38 +36892*x^37 -48989*x^36 -41768*x^35 +68397*x^34 +7921*x^33 +9893*x^32 -30841*x^31 +15927*x^30 +54995*x^29 +5474*x^28 -24546*x^27 -1559*x^26 -11350*x^25 +6196*x^24 -2886*x^23 -761*x^22 -3634*x^21 -13769*x^20 -6060*x^19 +880*x^18 +1445*x^17 -702*x^16 -1515*x^15 -1843*x^14 -223*x^13 -511*x^12 +172*x^11 +399*x^10 -153*x^9 -198*x^8 -61*x^7 +19*x^6 +21*x^5 +16*x^4 +7*x^3 +4*x^2 -1).

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.

A349187 Number of tilings of a 5 X n rectangle using n pentominoes of shapes X, Y, Z.

Original entry on oeis.org

1, 0, 0, 0, 0, 6, 6, 6, 2, 10, 86, 118, 166, 152, 372, 1394, 2450, 3866, 4946, 10160, 26380, 50770, 86522, 131632, 251150, 548436, 1075036, 1918294, 3205242, 5953962, 11962044, 23255472, 42565706, 74859582, 138078796, 266506794, 511327170, 947685504, 1713749022

Offset: 0

Alois P. Heinz, Nov 09 2021

			a(5) = 6:
  ._________.     ._________.
  |_. ._._| |     | |___. ._|
  | |_| |_. |     | |_  |_| |
  | |_. ._| |     | ._| |_. |
  | ._|_| |_| (2) |_| |___| | (4)
  |_|_______|     |_______|_|      .
.
a(8) = 2:
  ._______________.
  |_. | |___. ._| |
  | | |___. |_|_. |
  | |___| |_|_. | |
  | ._| |___. | |_| (2)
  |_|_______|_|___|      .
.

Alois P. Heinz, Table of n, a(n) for n = 0..3700
Wikipedia, Pentomino
Index entries for linear recurrences with constant coefficients, signature (0,0,0,3,13,7,8,1,-10,-14,-20,-8,-7,-38,-33,-2,0,-5,-16,-11).

Cf. A174249, A343529.

G.f.: (x^20 +6*x^19 +5*x^18 +7*x^15 +14*x^14 +7*x^13 +4*x^10 +2*x^9 +x^8 -2*x^7 -x^6 -7*x^5 -3*x^4 +1) / (11*x^20 +16*x^19 +5*x^18 +2*x^16 +33*x^15 +38*x^14 +7*x^13 +8*x^12 +20*x^11 +14*x^10 +10*x^9 -x^8 -8*x^7 -7*x^6 -13*x^5 -3*x^4 +1).

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.

cp's OEIS Frontend

A247712 Number T(n,k) of tilings of a 5 X n rectangle with pentominoes of any shape and exactly k pentominoes of shape Y; triangle T(n,k), n>=0, read by rows.

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A247713 Number T(n,k) of tilings of a 5 X n rectangle with pentominoes of any shape and exactly k pentominoes of shape Z; triangle T(n,k), n>=0, read by rows.

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

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A343529 Number of tilings of a 5 X n rectangle using n pentominoes of shapes P, X, Y.

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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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A349187 Number of tilings of a 5 X n rectangle using n pentominoes of shapes X, Y, Z.

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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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