cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A354011 Number of tilings of a 3 X n rectangle using 2 X 2 tiles, right trominoes and dominoes.

Original entry on oeis.org

1, 0, 7, 8, 81, 184, 1051, 3176, 14609, 50408, 210903, 773888, 3102369, 11711856, 46045259, 176114128, 686258465, 2640610128, 10247733223, 39540368248, 153162778865, 591718044968, 2290106238779, 8852558325048, 34248315785777, 132424316290104, 512224146701367
Offset: 0

Views

Author

Gerhard Kirchner, May 14 2022

Keywords

Comments

For tiling algorithm, see A351322.

Examples

			a(2)=7:
   ___    ___    ___    ___    ___    ___    ___
  |   |  |___|  |_  |  |  _|  |___|  |___|  |_|_|
  |___|  |   |  | |_|  |_| |  |___|  |_|_|  |_|_|
  |___|  |___|  |___|  |___|  |___|  |_|_|  |___|
		

Crossrefs

Programs

Formula

G.f.: (1 - 2*x - 2*x^2 + 2*x^3 - x^4) / (1 - 2*x - 9*x^2 + 8*x^3 - 3*x^4 - 6*x^5 + 3*x^6).
a(n)=2*a(n-1) + 9*a(n-2) - 8*a(n-3) + 3*a(n-4) + 6*a(n-5) - 3*a(n-6).

A354012 Number of tilings of a 4 X n rectangle using 2 X 2 tiles, right trominoes and dominoes.

Original entry on oeis.org

1, 1, 17, 81, 702, 4623, 35044, 248045, 1819731, 13110984, 95362462, 690253391, 5008926698, 36300216768, 263252448712, 1908449014617, 13837881924141, 100326715619679, 727420462629671, 5274035027493046, 38238994112367061, 277246970248002472, 2010151423463689959
Offset: 0

Views

Author

Gerhard Kirchner, May 14 2022

Keywords

Comments

For tiling algorithm, see A351322.

Examples

			a(2)=17, mirroring included (h: horizontal, v: vertical):
    v     v          h,v                      v           h
   ___   ___   ___   ___   ___   ___   ___   ___   ___   ___
  |   | |   | |   | |___| |___| | | | |___| |___| |___| |  _|
  |___| |___| |___| |_  | |___| |_|_| | | | |___| |   | |_| |
  |___| | | | |   | | |_| |___| | | | |_|_| | | | |___| | |_|
  |___| |_|_| |___| |___| |___| |_|_| |___| |_|_| |___| |___|
    2  +  2  +  1  +  4  +  1  +  1  +  1  +  2  +  1  +  2  = 17.
		

Crossrefs

Programs

Formula

G.f.: (1 - 4*x - 16*x^2 + 37*x^3 + 32*x^4 - 34*x^5 + 4*x^6 + 2*x^7 - 2*x^8) / (1 - 5*x - 28*x^2 + 69*x^3 + 142*x^4 - 194*x^5 - 78*x^6 + 57*x^7 + 36*x^8 - 70*x^9 + 32*x^10).
a(n)=5*a(n-1) + 28*a(n-2) - 69*a(n-3) - 142*a(n-4) + 194*a(n-5) + 78*a(n-6) - 57*a(n-7) - 36*a(n-8) + 70*a(n-9) - 32*a(n-10).

A354119 Number of tilings of an n X n square using right trominoes, dominoes, and 2 X 2 tiles.

Original entry on oeis.org

1, 0, 3, 8, 702, 41952, 16600824, 13298557992, 43157780553934, 388059824658319728, 12013678896331086695556, 1135287009899570278101009440, 350469343280046907242548361992474, 340321401792584726949313213285940152032, 1061786528674829112355995668763159293494203380
Offset: 0

Views

Author

Alois P. Heinz, May 17 2022

Keywords

Examples

			a(2) = 3:
   .___.    .___.    .___.
   |   |    |___|    | | |
   |___|    |___|    |_|_|  .
a(3) = 8:
  ._____.  ._____.  ._____.  ._____.
  |___| |  |_. | |  |___| |  |___| |
  | ._|_|  | |_|_|  | |___|  | | |_|
  |_|___|  |_|___|  |_|___|  |_|___|
  ._____.  ._____.  ._____.  ._____.
  | |___|  | |___|  | |___|  | | ._|
  |___| |  |_| | |  |_|_. |  |_|_| |
  |___|_|  |___|_|  |___|_|  |___|_|  .
		

Crossrefs

Main diagonal of A354010.
Cf. A219994.

Formula

a(n) = A354010(n,n).
Showing 1-3 of 3 results.