A247125 Number of tilings of a 5 X n rectangle using n pentominoes of shapes L, U, X.

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

0,3

Links

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

Wikipedia, Pentomino

Index entries for linear recurrences with constant coefficients, signature (0,2,1,12,6,2).

Formula

G.f.: -1/(2*x^6+6*x^5+12*x^4+x^3+2*x^2-1).

Example

a(4) = 16:
._______.     ._______.     ._______.
| ._____|     | ._____|     | ._| ._|
|_| |_. |     |_| |_. |     | | | | |
|_. ._| |     |_. ._| |     | | | | |
| |_|___|     | |_| | |     |_| |_| |
|_______| (2) |_____|_| (4) |___|___| (4)
._______.     ._______.
| ._____|     | ._____|
|_| ._. |     |_|_. | |
| |_| |_|     | ._| | |
|_____| |     | |___| |
|_______| (2) |___|___| (4) .

Maple

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);

Mathematica

LinearRecurrence[{0, 2, 1, 12, 6, 2}, {1, 0, 2, 1, 16, 10}, 35] (* Paolo Xausa, May 18 2026 *)

Prog

(PARI) 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*[1; 0; 2; 1; 16; 10])[1, 1] \\ Charles R Greathouse IV, May 15 2026

Crossrefs

Cf. A174249, A233427, A234312, A247124, A247268.

Sequence in context: A095850 A324610 A373034 * A290315 A113108 A381670

Adjacent sequences: A247122 A247123 A247124 * A247126 A247127 A247128

Keyword

nonn,easy

Author

Alois P. Heinz, Nov 19 2014

Status

approved