A247125
Number of tilings of a 5 X n rectangle using n pentominoes of shapes L, U, X.
Original entry on oeis.org
1, 0, 2, 1, 16, 10, 59, 60, 330, 397, 1520, 2218, 7875, 12820, 39250, 70045, 202168, 384866, 1038051, 2073580, 5385754, 11156701, 28015232, 59580154, 146333795, 317517636, 766142242, 1686735709, 4019319048, 8946988370, 21116854115, 47386013020, 111065223914
Offset: 0
a(4) = 16:
._______. ._______. ._______.
| ._____| | ._____| | ._| ._|
|_| |_. | |_| |_. | | | | | |
|_. ._| | |_. ._| | | | | | |
| |_|___| | |_| | | |_| |_| |
|_______| (2) |_____|_| (4) |___|___| (4)
._______. ._______.
| ._____| | ._____|
|_| ._. | |_|_. | |
| |_| |_| | ._| | |
|_____| | | |___| |
|_______| (2) |___|___| (4) .
-
a:= n-> (<<0|1|0|0|0|0>, <0|0|1|0|0|0>, <0|0|0|1|0|0>,
<0|0|0|0|1|0>, <0|0|0|0|0|1>, <2|6|12|1|2|0>>^n)[6,6]:
seq(a(n), n=0..40);
A247268
Number of tilings of a 5 X n rectangle using n pentominoes of shapes Y, U, X.
Original entry on oeis.org
1, 0, 0, 1, 0, 2, 1, 0, 4, 5, 38, 22, 13, 90, 144, 457, 408, 386, 1267, 2230, 5912, 6481, 7098, 18896, 35433, 79634, 101232, 127501, 288304, 546652, 1113907, 1560356, 2148298, 4408181, 8335234, 15954116, 23827541, 35011426, 67591204, 126376945, 232719926
Offset: 0
a(3) = 1, a(5) = 2:
._____. ._________. ._________.
| ._. | |_. .___| | | |___. ._|
|_| |_| | |_| |_. | | ._| |_| |
|_. ._| , | |_. ._| | | |_. ._| |
| |_| | | ._|_| |_| |_| |_|_. |
|_____| |_|_______| |_______|_| .
-
gf:= -(x^40 +12*x^39 +36*x^38 -5*x^36 -2*x^35 +12*x^34 +54*x^33 +4*x^32 -21*x^31 -23*x^30 +4*x^29 +20*x^28 +4*x^27 -4*x^25 -7*x^24 -6*x^23 -3*x^22 +33*x^21 -7*x^20 -10*x^19 -12*x^18 -9*x^17 +12*x^16 +16*x^15 +3*x^14 -2*x^13 -2*x^12 -2*x^11 -3*x^10 +5*x^9 -2*x^6 -7*x^5 -x^4 +1) /
(x^43 +12*x^42 +36*x^41 -3*x^40 -29*x^39 -58*x^38 +12*x^37 +67*x^36 +4*x^35 -123*x^34 -99*x^33 +8*x^32 +23*x^31 -145*x^30 -52*x^29 -52*x^28 -35*x^27 -112*x^26 -99*x^25 -28*x^24 -7*x^23 -15*x^22 -99*x^21 -42*x^20 +22*x^19 +36*x^18 +26*x^17 -4*x^16 +6*x^15 +31*x^14 +5*x^13 +11*x^12 +14*x^11 +23*x^10 -5*x^9 -7*x^8 -x^7 +2*x^6 +9*x^5 +x^4 +x^3 -1):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..60);
A264812
Number of tilings of a 5 X n rectangle using n pentominoes of shapes P, I, X.
Original entry on oeis.org
1, 1, 3, 5, 13, 52, 123, 366, 909, 2444, 7108, 19157, 53957, 146826, 400704, 1115852, 3059907, 8475420, 23369304, 64225984, 177572352, 488839323, 1349102071, 3722419367, 10255126169, 28303059509, 78013005366, 215160477217, 593488173404, 1636220978049
Offset: 0
a(4) = 13:
._______. ._______. ._______. ._______.
| | | | | | | | | | | | | ._| |
| | | | | | ._| ._| | ._| | | |___| |
| | | | | |_| |_| | |_| | | | | |___|
| | | | | (1) | | | (4) | | | | (6) | ._| | (2)
|_|_|_|_| |___|___| |_ _|_|_| |_|_____| .
a(5) = 52:
._________.
| |_. |
| ._| |___|
|_|_ _| |
| |_| | (2) ...
|_____|___| .
A278330
Number of tilings of a 5 X n rectangle using n pentominoes of shapes P, U, X.
Original entry on oeis.org
1, 0, 2, 1, 12, 10, 59, 52, 276, 349, 1404, 1984, 7019, 11148, 35686, 62181, 182776, 339350, 942507, 1841208, 4887096, 9921685, 25442304, 53190380, 132928715, 284198328, 696276202, 1514363221, 3654567764, 8053235650, 19212546163, 42762014028, 101125071372
Offset: 0
a(2) = 2, a(3) = 1:
.___. .___. ._____.
| | | | | ._. |
| ._| |_. | |_| |_|
|_| | | |_| |_ _|
| | | | | |_| |
|___| |___| |_____| .
- Alois P. Heinz, Table of n, a(n) for n = 0..1000
- Wikipedia, Pentomino
- Index entries for linear recurrences with constant coefficients, signature (0,2,2,8,4,21,-8,-4,-6,0,-16,-8).
Cf.
A079978,
A174249,
A233427,
A234312,
A234931,
A247124,
A247268,
A247443,
A249762,
A264765,
A264812.
-
a:= n-> (Matrix(12, (i, j)-> `if`(i+1=j, 1, `if`(i=12,
[-8, -16, 0, -6, -4, -8, 21, 4, 8, 2, 2, 0][j], 0)))^n.
<<1, 0, 2, 1, 12, 10, 59, 52, 276, 349, 1404, 1984>>)[1, 1]:
seq(a(n), n=0..35);
Showing 1-4 of 4 results.