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 1-5 of 5 results.

A233320 Number A(n,k) of tilings of a k X n rectangle using trominoes of any shape; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 3, 3, 0, 1, 1, 0, 0, 10, 0, 0, 1, 1, 1, 0, 23, 23, 0, 1, 1, 1, 0, 11, 62, 0, 62, 11, 0, 1, 1, 0, 0, 170, 0, 0, 170, 0, 0, 1, 1, 1, 0, 441, 939, 0, 939, 441, 0, 1, 1, 1, 0, 41, 1173, 0, 8342, 8342, 0, 1173, 41, 0, 1
Offset: 0

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Author

Alois P. Heinz, Dec 07 2013

Keywords

Comments

Every row and column satisfies a linear recurrence. - Peter Kagey, Jul 17 2019

Examples

			Square array A(n,k) begins:
  1, 1,  1,    1,   1,    1,       1, ...
  1, 0,  0,    1,   0,    0,       1, ...
  1, 0,  0,    3,   0,    0,      11, ...
  1, 1,  3,   10,  23,   62,     170, ...
  1, 0,  0,   23,   0,    0,     939, ...
  1, 0,  0,   62,   0,    0,    8342, ...
  1, 1, 11,  170, 939, 8342,   80092, ...
  1, 0,  0,  441,   0,    0,  614581, ...
  1, 0,  0, 1173,   0,    0, 5271923, ...

Links

Crossrefs

Columns (or rows) include: A000012, A001835, A134438, A233339, A233340, A233290, A233343, A269958, A215826, A269959, A269664.
Cf. A099390, A230031, A233427, A270061, A364457.

Formula

A(n,k) = 0 <=> n*k mod 3 > 0.

A134438 Number of tilings of a 3 X n rectangle with n trominoes.

Original entry on oeis.org

1, 1, 3, 10, 23, 62, 170, 441, 1173, 3127, 8266, 21937, 58234, 154390, 409573, 1086567, 2882021, 7645046, 20279829, 53794224, 142696606, 378522507, 1004078871, 2663452699, 7065162260, 18741269167, 49713692146, 131872134232, 349808216915, 927912454723
Offset: 0

Views

Author

Philippe Deléham, Jan 18 2008

Keywords

References

  • G. Kreweras, Recouvrements d'un rectangle de largeur 3 à l'aide de triminos, Mathématiques et sciences humaines, tome 130 (1995), p. 27-31.

Links

Crossrefs

Cf. A174248, A174249, A174250, A174251, A174252, A174253, A215826, A233290, A269664, A270063.
Column k=3 of A233320.

Programs

  • Maple
    a:= n-> (Matrix([[1$2, 0$2, 1, 0]]). Matrix(6, (i,j)-> if i+1=j then 1 elif j=1 then [1, 2, 6, 1, 0, -1][i] else 0 fi)^n)[1,2]: seq(a(n), n=0..30);  # Alois P. Heinz, Oct 09 2008
  • Mathematica
    LinearRecurrence[{1,2,6,1,0,-1},{1,1,3,10,23,62},40] (* Harvey P. Dale, Aug 27 2013 *)

Formula

a(n) = a(n-1) +2*a(n-2) +6*a(n-3) +a(n-4) -a(n-6).
G.f.: (1-x^3) / (1-x-2*x^2-6*x^3-x^4+x^6). - Alois P. Heinz, Oct 09 2008

Extensions

More terms from Alois P. Heinz, Oct 09 2008

A215826 Number of ways in which a 9 X n grid can be tiled with trominoes.

Original entry on oeis.org

1, 1, 41, 3127, 41813, 1269900, 45832761, 1064557805, 30860212081, 928789262080, 25020222581494, 714819627084057, 20574308184277971, 576115800837801057, 16381774291037991059, 466431115279461257920, 13190758349044182698371, 374524994697062170913555
Offset: 0

Views

Author

V. Raman, Aug 23 2012

Keywords

Links

Crossrefs

Cf. A215827 (number of memoizations needed to calculate a(n)), A134438, A233290, A269664.
Column k=9 of A233320.

A269664 Number of tilings of a 12 X n rectangle with 4n trominoes of any shape.

Original entry on oeis.org

1, 1, 153, 58234, 1895145, 198253934, 27438555522, 1949314526229, 193553900967497, 20574308184277971, 1830607857363940042, 178792253082742021463, 17735061025562799941630, 1679378707647721857218932, 163105210594579645492072521, 15894545877032388610890500803
Offset: 0

Views

Author

Alois P. Heinz, Mar 02 2016

Keywords

Links

Crossrefs

Column k=12 of A233320.
Cf. A134438, A215826, A233290.

A351323 Number of tilings of a 6 X n rectangle with right trominoes.

Original entry on oeis.org

1, 0, 4, 8, 18, 72, 162, 520, 1514, 4312, 13242, 39088, 118586, 361712, 1103946, 3403624, 10513130, 32614696, 101530170, 316770752, 990771834, 3104283168, 9741133578, 30606719000, 96263812906, 303028237848, 954563802106, 3008665176560, 9487377712634, 29928407213328
Offset: 0

Views

Author

Gerhard Kirchner, Feb 21 2022

Keywords

Comments

See A351322 for algorithm. The subsequence 1,8,162,... for 6 X 3n rectangles also has a depending recurrence with 11 parameters.
The sequence is the Hadamard sum of the following 4 sequences: 0, 0, 0, 0, 16, 0, 128, 0, 256, 768, 1024,0, 13440, 0, 16384, .. (tilings which have both horizontal and vertical faults), 0, 0, 4, 8, 0, 0, 16, 0, 0, 128, 0, 0, 1536, 0, 0,.. (tilings which have horizontal faults but no vertical faults), 0, 0, 0, 0, 0, 64, 16, 480, 1140, 3200, 11208, 36032, 95924, 333856, 1003096,.. (tilings which have vertical faults but no horizontal faults), 1, 0, 0, 0, 2, 8, 2, 40, 118, 216, 1010, 3056, 7686, 27856, 84466,... (tilings which have neither vertical nor horizontal faults). - R. J. Mathar, Dec 08 2022

Examples

			For a 6 X 2 rectangle there are 4 tilings:
   ___   ___   ___   ___
  |  _| |  _| |_  | |_  |
  |_| | |_| | | |_| | |_|
  |___| |___| |___| |___|
  |  _| |_  | |  _| |_  |
  |_| | | |_| |_| | | |_|
  |___| |___| |___| |___|

Links

Crossrefs

Cf. A077957, A000079, A046984, A084478, A351322, A351324, A236576 (straight trominoes), A233290 (mixed trominoes).

Formula

G.f.: (1 - x)*(1 - x - 5*x^2 - 7*x^3 + 6*x^4 + 12*x^5 + 6*x^6)/(1 - 2*x - 8*x^2 - 2*x^3 + 43*x^4 + 42*x^5 - 36*x^6 - 102*x^7 + 44*x^9 + 8*x^10 + 8*x^11).
a(n) = Sum_{i=0..10} b(i)*a(n-11+i) for n>10 where {b(i)} = {-8,-8,-44,0,102,36,-42,-43,2,8,2}.
Showing 1-5 of 5 results.

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