A247076 -id:A247076 - cp's OEIS frontend

A247706 Number T(n,k) of tilings of a 5 X n rectangle with pentominoes of any shape and exactly k pentominoes of shape P; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 1, 0, 3, 0, 2, 16, 20, 20, 0, 135, 204, 140, 16, 6, 944, 1432, 1164, 296, 170, 0, 4814, 8796, 8452, 4068, 1708, 92, 20, 26435, 58656, 66994, 41648, 17494, 2700, 762, 0, 158761, 410000, 520728, 371456, 175810, 46648, 12876, 440, 62, 978044, 2783560, 3836254, 3107308, 1696312, 609772, 172724, 18220, 3160, 0

Offset: 0

Alois P. Heinz, Sep 22 2014

Sum_{k>0} k * T(n,k) = A247739(n).

			T(2,2) = 2:
.___.   .___.
|   |   |   |
| ._|   |_. |
|_| |   | |_|
|   |   |   |
|___|   |___| .
Triangle T(n,k) begins:
00 :      1;
01 :      1,      0;
02 :      3,      0,      2;
03 :     16,     20,     20,      0;
04 :    135,    204,    140,     16,      6;
05 :    944,   1432,   1164,    296,    170,     0;
06 :   4814,   8796,   8452,   4068,   1708,    92,    20;
07 :  26435,  58656,  66994,  41648,  17494,  2700,   762,   0;
08 : 158761, 410000, 520728, 371456, 175810, 46648, 12876, 440, 62;

Alois P. Heinz, Rows n = 0..140, flattened
Wikipedia, Pentomino

Row sums give A174249 or A233427(n,5).
Column k=0 gives A247770.
Even bisection of main diagonal gives A247076.
Cf. A247739.

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).

cp's OEIS Frontend

A247706 Number T(n,k) of tilings of a 5 X n rectangle with pentominoes of any shape and exactly k pentominoes of shape P; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs