A349187 -id:A349187 - cp's OEIS frontend

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

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);

A358933 Number of tilings of a 5 X n rectangle using n pentominoes of shapes N, U, Z.

Original entry on oeis.org

1, 0, 0, 0, 2, 0, 2, 2, 4, 2, 10, 8, 14, 18, 36, 34, 66, 88, 136, 170, 292, 382, 578, 818, 1244, 1692, 2576, 3676, 5400, 7654, 11412, 16284, 23852, 34448, 50396, 72472, 106046, 153556, 223458, 323430, 471644, 683046, 993958, 1442138, 2097830, 3042314, 4424880

Offset: 0

Alois P. Heinz, Dec 06 2022

			a(7) = 2:
  ._____________.    ._____________.
  | | ._. | ._. |    | ._. | ._. | |
  | |_| |_|_| |_|    |_| |_|_| |_| |
  |_. |___. |_. |    | ._| .___| ._|
  | |_| | |_| | |    | | |_| | |_| |
  |_____|_____|_|    |_|_____|_____|   .
.

Alois P. Heinz, Table of n, a(n) for n = 0..2000
Wikipedia, Pentomino
Index entries for linear recurrences with constant coefficients, signature (0, 0, 5, 2, 10, -4, -7, -35, -26, -34, 13, 43, 94, 115, 51, 30, -124, -99, -276, -52, -131, 182, 128, 298, 189, 71, -30, -118, -118, -96, -56, -8).

Cf. A174249, A343529, A349187, A352421, A361250.

G.f.: (x-1)*(x^2 +x +1)*(4*x^26 +28*x^25 +44*x^24 +29*x^23 +x^22 -36*x^21 -49*x^20 -45*x^19 -61*x^18 +15*x^17 -7*x^16 +60*x^15 +x^14 +59*x^13 -11*x^12 +10*x^11 -37*x^10 -25*x^8 +x^7 -2*x^6 +10*x^5 +4*x^3 -1) / (8*x^32 +56*x^31 +96*x^30 +118*x^29 +118*x^28 +30*x^27 -71*x^26 -189*x^25 -298*x^24 -128*x^23 -182*x^22 +131*x^21 +52*x^20 +276*x^19 +99*x^18 +124*x^17 -30*x^16 -51*x^15 -115*x^14 -94*x^13 -43*x^12 -13*x^11 +34*x^10 +26*x^9 +35*x^8 +7*x^7 +4*x^6 -10*x^5 -2*x^4 -5*x^3 +1).

cp's OEIS Frontend

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

A358933 Number of tilings of a 5 X n rectangle using n pentominoes of shapes N, U, Z.

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