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.

A353877 Triangle read by rows: T(n,k) = number of tilings of a n X k rectangle using right trominoes, dominoes and 1 X 1 tiles, n >= 0, k = 0..n.

Original entry on oeis.org

1, 1, 1, 1, 2, 11, 1, 3, 44, 369, 1, 5, 189, 3633, 83374, 1, 8, 798, 34002, 1817897, 90916452, 1, 13, 3383, 323293, 40220893, 4635661331, 546063639624, 1, 21, 14328, 3058623, 886130549, 235025597912, 63919977468729, 17259079054003609, 1, 34, 60697, 28982628, 19546906987, 11935601703140, 7495901454256347, 4669873251135795702, 2916019543694306398589
Offset: 0

Views

Author

Gerhard Kirchner, May 09 2022

Keywords

Comments

Tiling algorithm, see A351322.
Reading the sequence {T(n,k)} for k>n, use T(k,n) instead of T(n,k).

Examples

			Triangle begins
n\k_0__1____2______3________4__________5____________6
0:  1
1:  1  1
2:  1  2   11
3:  1  3   44    369
4:  1  5  189   3633    83374
5:  1  8  798  34002  1817897   90916452
6:  1 13 3383 323293 40220893 4635661331 546063639624

Links

Crossrefs

Row/columns 0..4 are A000012, A000045(n+1), A110679, A353878, A353879.
Main diagonal is A353934.
Cf. A219987, A351322, A352589.

Programs

A353879 Number of tilings of a 4 X n rectangle using right trominoes, dominoes and 1 X 1 tiles.

Original entry on oeis.org

1, 5, 189, 3633, 83374, 1817897, 40220893, 886130549, 19546906987, 431024540644, 9505433227293, 209617856008535, 4622624792880217, 101940750143038657, 2248057208102711472, 49575464007447758483, 1093267021618939507743, 24109360928450426884813, 531673668551361276666101
Offset: 0

Views

Author

Gerhard Kirchner, May 09 2022

Keywords

Comments

For tiling algorithm see A351322.

Examples

			a(2)=189.
The number of tilings (mirroring included) using r trominoes
      ___   ___   ___   ___
r=1: |  _| |  _| | |_| |_2_|    r=0: 71 = A030186(4)
     |_|_| |_| | |___| |_  |
     | 7 | |3|_| | 7 | |3|_|
     |___| |___| |___| |___|
      4*7 + 4*3 + 4*7 + 4*6 = 92
      ___   ___   ___   ___   ___   ___   ___
r=2: |  _| |  _| |  _| |  _| |  _| | |_| | |_|
     |_| | |_|2| |_|_| |_|_| |_|_| |___| |___|
     |___| | |_| |  _|_|_| | |_  | |_  | |  _|
     |_2_| |___| |_|_| |___| |_|_| |_|_| |_|_|
      4*2 + 2*2 + 4*1 + 2*1 + 4*1 + 2*1 + 2*1 = 26
Result: a(2) = 71+92+26 = 189.
Legend:
   ___              ___      ___
  |_2_| stands for |___| or |_|_|
     _                _        _        _
   _|3|             _| |     _|_|     _|_|
  |___| stands for |_|_| or |___| or |_|_|
   ___              ___   ___   ___   ___   ___   ___      ___
  | 7 |            |___| |_|_| |___| | | | |_| | | |_|    |_|_|
  |___| stands for |___|,|___|,|_|_|,|_|_|,|_|_|,|_|_| or |_|_|

Links

Crossrefs

Cf. A030186, A127870, A165791, A351322, A352589, A353877, A353878.

Programs

Formula

G.f.: (1 - 9*x - 64*x^2 + 109*x^3 + 39*x^4 + 41*x^5 + 12*x^6 - 7*x^7 - 2*x^8) / (1 - 14*x - 183*x^2 + 37*x^3 + 1929*x^4 - 2419*x^5 + 212*x^6 + 333*x^7 - 25*x^8-15*x^9).
a(n) = 14*a(n-1) + 183*a(n-2) - 37*a(n-3) - 1929*a(n-4) + 2419*a(n-5) - 212*a(n-6) - 333*a(n-7) + 25*a(n-8) + 15*a(n-9).

A354131 Number of tilings of a 2 X n rectangle using 2 X 2 and 1 X 1 tiles, right trominoes and dominoes.

Original entry on oeis.org

1, 2, 12, 48, 216, 936, 4104, 17928, 78408, 342792, 1498824, 6553224, 28652616, 125277192, 547747272, 2394904968, 10471198536, 45783025416, 200176267464, 875226954888, 3826738469448, 16731577137672, 73155162229704, 319854949515144, 1398495821923656
Offset: 0

Views

Author

Gerhard Kirchner, May 18 2022

Keywords

Comments

Tiling algorithm see A351322.

Examples

			a(3)=48
Number of tilings without a 2 X 2 square: 44, see A353878.
Number of other tilings: 4
   ___ _   ___ _   _ ___   _ ___
  |   | | |   |_| | |   | |_|   |
  |___|_| |___|_| |_|___| |_|___|

Links

Crossrefs

Cf. A351322, A352589, A353878, A354130, A354132.

Programs

Formula

G.f.: (1 - x) / (1 - 3*x - 6*x^2).
a(n) = 3*a(n-1) + 6*a(n-2).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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